PARAMETRIC DYNAMIC LOAD PREDICTION OF A NARROW … · PARAMETRIC DYNAMIC LOAD PREDICTION OF A...
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PARAMETRIC DYNAMIC LOAD PREDICTION OF A NARROW GAUGE ROCKET SLED BY JOHN SCOTT FURLOW, B.S., M.S. A dissertation submitted to the Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy, Engineering Specialization in: Mechanical Engineering New Mexico State University Las Cruces, New Mexico December 2006
Transcript of PARAMETRIC DYNAMIC LOAD PREDICTION OF A NARROW … · PARAMETRIC DYNAMIC LOAD PREDICTION OF A...
PARAMETRIC DYNAMIC LOAD PREDICTION OF A NARROW GAUGE
ROCKET SLED
BY
JOHN SCOTT FURLOW, B.S., M.S.
A dissertation submitted to the Graduate School
in partial fulfillment of the requirements
for the degree
Doctor of Philosophy, Engineering
Specialization in: Mechanical Engineering
New Mexico State University
Las Cruces, New Mexico
December 2006
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DEDICATION
This work is dedicated to all my family most especially my wife, Debbie, and
children, Steven and Morgan, as they were the ones who truly sacrificed for the
completion of this work. This work is also dedicated to the promise of God as: ‘No
eye has seen, no ear has heard, no mind has conceived what God has prepared for
those who love Him.’ 1 Cor 2:9.
v
ACKNOWLEDGEMENTS
I would like to acknowledge my advisor Dr. Gabe Garcia for his direction and
guidance in completing this degree and Dr. Michael Hooser for his direction and
guidance in the areas of data comparison and DADS modeling. I would also like to
thank Dr. Young H. Park, Dr. Ou Ma, and Dr. David V. Jáuregui for serving on my
Committee. Much appreciation and gratitude is given to the leadership of the 846TS
and 46TG for the opportunity to pursue this work. I would also like to acknowledge
Mr. Matthew Neidigk for his assistance in producing useful PRONTO results and Mr.
Dennis Turnbull for dissertation quality control. Mr. Cody McFarland is due many
thanks for work on the implementation of the multivariate interpolator in Excel.
I would like to acknowledge the support and patience of my family during the
completion of this work while enduring my plain sight absence from them during
much of it.
This work was supported in part by a grant of computer time from the DOD
High Performance Computing Modernization Program at ASC and ERDC.
vi
VITA
March 19, 1975 Born in Lubbock, Texas
1993 Graduated from Brownfield High School,
Brownfield, Texas
1997 Bachelor of Science, Mechanical Engineering,
Texas Tech University
1999 Master of Science, Mechanical Engineering,
Texas Tech University
1999-Present Mechanical Engineer, United States Air Force,
Holloman Air Force Base, New Mexico
PROFESSIONAL AND HONORARY SOCIETIES
AIAA
Tau Beta Pi
Pi Tau Sigma
vii
PUBLICATIONS
Furlow, J. S. and James, D. L., Convective Heat Transfer Characteristics from
Combined Mechanical and Supply Pulsed Radial Reattaching Jets, International
Journal of Heat and Fluid Flow, Accepted for Publication 8 February 2006.
Furlow, J.S. and James, D.L., Local Instantaneous Convective Heat Transfer
Characteristics of Radial Reattaching Nozzles, 33rd National Heat Transfer
Conference, Albuquerque, NM, NHTC99-151, August, 1999.
viii
ABSTRACT
PARAMETRIC DYNAMIC LOAD PREDICTION OF A NARROW GAUGE
ROCKET SLED
BY
JOHN SCOTT FURLOW, B.S., M.S.
Doctor of Philosophy, Engineering
Specialization in Mechanical Engineering
New Mexico State University
Las Cruces, New Mexico, 2006
Dr. Gabe V. Garcia, Chair
Dynamic load prediction of rocket sleds has been of interest to sled designers and
analysts since the inception of the Holloman High Speed Test Track, (HHSTT).
Dynamic loading along with thrust and aerodynamic loading is a primary contributor
to sled design load cases. Dynamic loading comes directly from the rocket sled
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traversing the gap between the slipper and rail and the resulting sliding impacts. The
current study investigates the prediction of narrow gauge sled dynamic loads by
applying a systematic process of modeling validation, design parameter variation and
dynamic load correlation.
Numerical modeling was employed to simulate the Land Speed Record (LSR) test
and the model data was validated by comparing it to the data taken from the sled
during the LSR test. Modeling methods validated against the test data were applied
to a reduced complexity narrow gauge sled representing a generic version of the LSR
sled. Design parameters were identified that contributed to the generation of dynamic
loading. The design parameters are: sled mass, slipper gap, vertical rail roughness,
lateral rail roughness, vertical sled natural frequency, lateral sled natural frequency,
torsional sled natural frequency, and sled velocity. Peak dynamic load results (from
evaluating the reduced complexity model while varying the design parameter values
over high, low, and typical ranges) were computed at the sled center of Gravity (CG).
This peak dynamic loading, η force, constituted the dynamic load prediction. The
correlation of η to its respective design parameters showed that a multivariate
interpolation method was the most accurate method to relate η force to its respective
design parameters. The study revealed a heavy dependence of dynamic load on
velocity, rail roughness, slipper gap, and translational sled natural frequencies. The
study also showed a favorable comparison of η force prediction over previously used
methods at the HHSTT.
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TABLE OF CONTENTS
LIST OF TABLES...................................................................................................... xv
LIST OF FIGURES ................................................................................................... xix
1. INTRODUCTION .................................................................................................... 1
1.1 Existing Theory................................................................................................... 1
1.1.1 Lambda, λ .................................................................................................... 3
1.1.2 SIMP/SLEDYNE......................................................................................... 5
1.1.3 Numerical Integration of Equations of Motion, NIEOM............................. 8
1.2 Limitations of Current Methods.......................................................................... 9
1.3 Problem Statement ............................................................................................ 10
2. DYNAMIC LOAD PREDICTION OF NARROW GAUGE SLEDS ................... 13
2.1 Narrow Gauge Sled Dynamic Load Prediction ................................................ 13
2.2 Selection of Dynamic Load Prediction Method................................................ 16
2.3 Development of Dynamic Load Prediction Method......................................... 17
2.3.1 Construction and Validation of DADS Model........................................... 17
2.3.2 Wide Scale Design Parameter Variation.................................................... 18
2.3.3 Correlation of Design Parameters to Dynamic Loading Outputs .............. 19
2.3.4 Evaluation of Design Tool Effectiveness Compared to Previous Methods20
3. DEVELOPMENT OF NARROW GAUGE SLED MODEL................................. 21
3.1 Modeling Assumptions ..................................................................................... 22
3.2 Design Parameter Selection .............................................................................. 22
3.3 Slipper Rail Impact Modeling........................................................................... 25
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3.3.1 Theoretical Formulation............................................................................. 29
3.3.2 Computational Formulation ....................................................................... 30
3.3.3 DADS Implementation .............................................................................. 36
3.4 Flexible Body Modeling ................................................................................... 38
3.4.1 Modal Representation ................................................................................ 45
3.4.2 Spring Mass Damper Representation......................................................... 50
3.5 Sled Model Formation ...................................................................................... 53
3.5.1 Three Dimensional Structural Effects........................................................ 54
3.5.2 Nonlinear Boundary Conditions ................................................................ 54
3.5.3 External Loading, Aerodynamic and Thrusting......................................... 66
3.5.4 Rocket Motor Mass Variation.................................................................... 71
4. EVALUATION OF NUMERICAL MODEL AND OUTPUT CHARACTERISTIC STUDY................................................................................................................... 73
4.1 Evaluation of Numerical Model........................................................................ 73
4.1.1 Time Domain Comparison......................................................................... 76
4.1.2 Frequency Domain Comparison ................................................................ 77
4.1.3 Correlation Criteria .................................................................................... 79
4.2 Output Characteristic Study.............................................................................. 81
4.2.1 Modal Model Characteristics ..................................................................... 81
4.2.2 SMD Model Characteristics....................................................................... 85
4.3 Flexible Body Representation Selection........................................................... 88
5. EVALUATION OF LARGE SCALE DESIGN PARAMETER VARIATION .... 92
5.1 Construction of Reduced Complexity Parameter Variation Model.................. 92
5.1.1 Sled Body................................................................................................... 93
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5.1.2 Slipper Beam.............................................................................................. 96
5.1.3 Slippers ...................................................................................................... 97
5.2 Design Parameter Formulation and Associated Values.................................... 97
5.2.1 Sled Mass ................................................................................................. 102
5.2.2 Slipper Gap .............................................................................................. 103
5.2.3 Rail Roughness ........................................................................................ 105
5.2.4 Sled Natural Frequency............................................................................ 105
5.2.5 Velocity.................................................................................................... 106
5.3 η Force Formation........................................................................................... 107
5.4 Design Parameter Variation............................................................................ 111
5.4.1 Initial Design Parameter Variation .......................................................... 115
5.4.2 Intermediate Design Parameter Variation................................................ 123
5.4.3 Final Design Parameter Variation............................................................ 123
6. DEVELOPMENT OF CORRELATION OF PREDICTED LOAD AND DESIGN PARAMETERS ................................................................................................... 126
6.1 Linear Least Squares Regression .................................................................... 127
6.2 Nonlinear Least Squares Regression .............................................................. 130
6.2.1 Polynomial Form ..................................................................................... 131
6.2.2 Power Form.............................................................................................. 133
6.3 Nondimensional Analysis ............................................................................... 134
6.4 Multivariate Interpolation ............................................................................... 140
7. COMPARISON OF PREDICTED LOAD TO PREVIOUS LOAD PREDICTION METHODS........................................................................................................... 147
7.1 Force Magnitude Comparison......................................................................... 147
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7.1.1 Vertical Force Comparison ...................................................................... 149
7.1.2 Lateral Force Comparison........................................................................ 151
7.2 Sled Design Comparison................................................................................. 152
7.2.1 λ Design Study......................................................................................... 157
7.2.2 SIMP Design Study.................................................................................. 160
7.2.3 η Design Study......................................................................................... 167
7.2.4 Comparison of Design Study Results ...................................................... 168
7.3 CO Test Data Comparison .............................................................................. 174
8. DISCUSSION OF RESULTS .............................................................................. 176
8.1 Major Findings................................................................................................ 176
8.1.1 λ and SIMP Limits of Applicability ........................................................ 176
8.1.2 Applicability of Dynamic Structural Modeling Approaches ................... 177
8.1.3 Slipper Rail Impact Characterization....................................................... 177
8.1.4 Importance of Modeling of Sled Nonlinearities ...................................... 178
8.1.5 Importance of Accurate Damping Modeling ........................................... 179
8.1.6 Utility of Figure of Merit (FOM)............................................................. 179
8.1.7 Identification of Appropriate Design Parameters and Corresponding Ranges ..................................................................................................... 180
8.1.8 Superiority of Multivariate Interpolation to Correlate Design Parameters to η Forces ................................................................................................... 180
8.1.9 Influence of Design Parameters on Dynamic Loading ............................ 181
8.2 Significance and Contribution of Current Study ............................................ 181
9. CONCLUSIONS................................................................................................... 183
9.1 Existing Theory Review of Dynamic Load Prediction................................... 183
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9.2 Numerical Model Development...................................................................... 184
9.2.1 Modeling Assumptions ............................................................................ 184
9.2.2 Slipper Rail Impact Results ..................................................................... 185
9.2.3 DADS Implementation ............................................................................ 185
9.3 Evaluation of Numerical Model...................................................................... 186
9.4 Evaluation of Large Scale Design Parameter Variation ................................. 187
9.5 Correlation of Predicted Load and Design Parameters................................... 188
9.6 Comparison to Previous Methods................................................................... 189
9.7 Future Work .................................................................................................... 190
9.8 Concluding Remarks....................................................................................... 192
Appendices
A. FEMS AND RESULTING COMPARISONS................................................. 193
B. FLEXIBLE BODY STIFFNESS FORMULATION ....................................... 204
C. MATLAB CORRELATION DATA PROCESSING SCRIPTS ..................... 208
D MATLAB PARAMETER VARIATION DATA PROCESSING SCRIPTS... 224
REFERENCES ......................................................................................................... 233
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LIST OF TABLES
Table 3.1 Modeling Assumptions Ensuring Model Fidelity....................................... 23
Table 3.2 Modeling Assumptions Outside Scope of Present Study ........................... 23
Table 3.3 PRONTO Slipper Impact Results ............................................................... 36
Table 3.4 DADS Slipper Impact Results, Linear Contact Model............................... 39
Table 3.5 DADS Slipper Impact Results, Nonlinear Contact Model ......................... 39
Table 3.6 DADS Slipper Impact Results, Hertzian Contact Model ........................... 39
Table 3.7 DADS Model Acceptance Criterion ........................................................... 42
Table 3.8 Major Structural Component Frequency Properties ................................... 44
Table 3.9 Major Structural Component Mass Properties............................................ 44
Table 3.10 Major Structural Component Static Deflection Properties ....................... 45
Table 3.11 Modal Model Empty Case Rocket Motor Mass Properties Comparison.. 49
Table 3.12 Modal Model Rocket Motor Frequency Comparison for Vibration Test, Mixon (2002b).......................................................................................... 49
Table 3.13 SMD Model Mass Properties.................................................................... 53
Table 3.14 SMD Model Frequency Properties ........................................................... 53
Table 3.15 Nonlinear Boundary Condition Summary ................................................ 58
Table 3.16 Instrumentation Pallet Foam Vibration Isolation Characteristics............. 63
Table 3.17 Instrumentation Pallet Rigid Mass Mass Properties ................................. 64
Table 3.18 Propellant Mass Properties ....................................................................... 72
Table 4.1 Time Domain Comparison Calculation ...................................................... 77
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Table 4.2 RMS Power Calculation ............................................................................. 80
Table 4.3 LSR Modal Model Output Measurement ................................................... 82
Table 4.4 CO Modal Model Output Measurement ..................................................... 82
Table 4.5 LSR SMD Model Output Measurement ..................................................... 86
Table 4.6 CO SMD Model Output Measurement....................................................... 87
Table 4.7 LSR Modal FOM Summary ....................................................................... 89
Table 4.8 LSR SMD FOM Summary ......................................................................... 90
Table 4.9 CO Modal FOM Summary ......................................................................... 90
Table 4.10 CO SMD FOM Summary ......................................................................... 91
Table 5.1 Sled Body Mass Distribution...................................................................... 95
Table 5.2 Sled Body Flexibility .................................................................................. 96
Table 5.3 Slipper Beam Mass Distribution................................................................. 98
Table 5.4 Slipper Beam Flexibility............................................................................. 99
Table 5.5 Slipper Mass Distribution ......................................................................... 102
Table 5.6 Slipper Flexibility ..................................................................................... 103
Table 5.7 Reduced Design Parameter Listing........................................................... 104
Table 5.8 Reduced Complexity Sled Mass Properties.............................................. 104
Table 5.9 Rail Roughness Standard Deviation Values ............................................. 105
Table 5.10 Reduced Complexity Model Thrust Values............................................ 107
Table 5.11 Design Parameter Combination Convention example............................ 114
Table 5.12 Intermediate Design Parameter Variation Values .................................. 123
Table 5.13 Final Design Parameter Variation Values .............................................. 125
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Table 6.1 Linear Multivariate Least Squares Regression Coefficients..................... 129
Table 6.2 Linear Multivariate Least Squares Correlation Coefficients .................... 130
Table 6.3 Polynomial Individual Correlation Coefficient Ranges............................ 132
Table 6.4 Power Form Exponent Values .................................................................. 134
Table 6.5 Power Form Individual Coefficient Values .............................................. 135
Table 6.6 Design Parameter Dimensional Analysis ................................................. 138
Table 6.7 Π Exponent Values................................................................................... 138
Table 6.8 Three Dimensional Interpolation Detail ................................................... 144
Table 7.1 Selected η Comparisons to λ and SIMP at 10,000 fps ............................. 149
Table 7.2 Maximum η Force Values at 10,000 fps .................................................. 149
Table 7.3 Slipper Beam Material Specification........................................................ 154
Table 7.4 Sled Body Material Specification ............................................................. 155
Table 7.5 Design Parameters for Mock Sled Design................................................ 155
Table 7.6 Initial Study Dimensions .......................................................................... 156
Table 7.7 Initial Study Weight Detail ....................................................................... 156
Table 7.8 λ Initial Design Parameters....................................................................... 158
Table 7.9 λ Final Study Dimensions......................................................................... 158
Table 7.10 SIMP Initial Design Parameters ............................................................. 161
Table 7.11 SIMP Final Design Parameters............................................................... 161
Table 7.12 SIMP Effective Mass Values.................................................................. 162
Table 7.13 SIMP Effective Impact Frequency ......................................................... 163
Table 7.14 SIMP Effective Impact Velocity............................................................. 163
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Table 7.15 SIMP Maximum Slipper Force............................................................... 164
Table 7.16 SIMP Final Study Dimensions ............................................................... 167
Table 7.17 η Final Design Parameters for Mock Sled Design ................................. 169
Table 7.18 η Final Study Dimensions ...................................................................... 169
Table 7.19 Slipper Beam Peak Moment Values ....................................................... 171
Table 7.20 Sled Body Peak Moment Values ............................................................ 171
Table 7.21 Final Design Study Results Comparison ................................................ 172
Table A.1 Modal Model Payload Mass Properties Comparison............................... 198
Table A.2 Modal Model Payload Frequency Comparison ....................................... 198
Table A.3 Modal Model Slipper Beam Mass Properties Comparison...................... 199
Table A.4 Modal Model Slipper Beam Frequency Comparison .............................. 199
Table A.5 Modal Model Slipper Mass Properties Comparison................................ 200
Table A.6 Modal Model Slipper Frequency Comparison......................................... 200
Table A.7 Spring Mass Damper Model Payload Mass Properties Comparison ....... 201
Table A.8 Spring Mass Damper Model Payload Frequency Comparison................ 201
Table A.9 Spring Mass Damper Model Slipper Beam Mass Properties Comparison ............................................................................ 202
Table A.10 Spring Mass Damper Model Slipper Beam Frequency Comparison..... 202
Table A.11 Spring Mass Damper Model Slipper Mass Properties Comparison ...... 203
Table A.12 Spring Mass Damper Model Slipper Frequency Comparison ............... 203
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LIST OF FIGURES
Figure 1.1 Slipper Rail Interaction................................................................................ 2
Figure 1.2 Hypersonic Narrow Gauge Sled.................................................................. 2
Figure 1.3 λ vs. Velocity with Measured Sled Data (Mixon and Hooser 2002) .......... 5
Figure 1.4 λ vs. Velocity............................................................................................... 6
Figure 1.5 Narrow Gauge λ Compared to Sled Tests................................................. 10
Figure 1.6 Monorail λ Compared to Sled Tests.......................................................... 11
Figure 2.1 Application of Vertical λ to a Narrow Gauge sled.................................... 14
Figure 3.1 LSR Forebody Sled ................................................................................... 22
Figure 3.2 Slipper Rail Impact Parallel Orientation ................................................... 26
Figure 3.3 Slipper Rail Impact Maximum Angled Orientation 3o.............................. 26
Figure 3.4 Front View of DADS Slipper Rail Contact Depiction .............................. 27
Figure 3.5 Isometric View of DADS Slipper Rail Contact ........................................ 27
Figure 3.6 Slipper Rail PRONTO Model Front View ................................................ 32
Figure 3.7 Slipper Rail PRONTO Model Side View.................................................. 33
Figure 3.8 Slipper Rail PRONTO Model Isometric View.......................................... 33
Figure 3.9 Slipper Rail PRONTO Model Boundary Conditions ................................ 34
Figure 3.10 Lateral Impact 100 ips, 5000 fps Sliding Velocity Comparison ............. 34
Figure 3.11 Vertical Up Impact 100 ips, 0 fps Sliding Velocity ................................ 35
Figure 3.12 Vertical Down Impact 50 ips, Linear Contact......................................... 40
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Figure 3.13 Vertical Down Impact 50 ips, Nonlinear Contact ................................... 40
Figure 3.14 Sled Structural Connection Depiction ..................................................... 55
Figure 3.15 LSR Velocity Profile ............................................................................... 56
Figure 3.16 CO Velocity Profile................................................................................. 56
Figure 3.17 Lateral Slipper Vibration Isolation.......................................................... 58
Figure 3.18 Nonlinear Polyurethane Force Deflection Test Data (Mixon 2002a) ..... 59
Figure 3.19 Aft Payload Downtrack and Lateral Vibration Isolation Mechanism..... 61
Figure 3.20 Gap Induced Nonlinear Stiffness at Payload Aft Vertical Attachment ... 61
Figure 3.21 Instrumentation Pallet Depiction............................................................. 63
Figure 3.22 Slipper Rail Interface Showing Slipper Gap ........................................... 64
Figure 3.23 C-Rail East Vertical Roughness, DADS Contact Points 1, 3, 4, 6.......... 66
Figure 3.24 C-Rail East Lateral Roughness, DADS Contact Point 2......................... 67
Figure 3.25 B-Rail West Lateral Roughness, DADS Contact Point 5........................ 67
Figure 3.26 B-Rail East Vertical Roughness, DADS Contact Points 4 and 6 ............ 68
Figure 3.27 B-Rail West Vertical Roughness, DADS Contact Points 1 and 3........... 68
Figure 3.28 LSR Component Drag ............................................................................. 69
Figure 3.29 LSR Component Lift ............................................................................... 69
Figure 3.30 LSR Rocket Motor Thrust ....................................................................... 70
Figure 3.31 CO Component Drag............................................................................... 70
Figure 3.32 CO Component Lift................................................................................. 71
Figure 4.1 LSR Data Channel Schematic ................................................................... 74
Figure 4.2 CO Data Channel Schematic ..................................................................... 74
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Figure 4.3 Time Domain Overlay, FOM 84.07%....................................................... 77
Figure 4.4 PSD Overlay, FOM 70.00%...................................................................... 79
Figure 5.1 Reduced Complexity Model Depiction ..................................................... 94
Figure 5.2 DADS Slipper Depiction of Front Right Slipper .................................... 100
Figure 5.3 Force Depiction of η Forces from Reduced Complexity Model............. 109
Figure 5.4 Envelope Function Depiction Vertical All-Typical Data Set.................. 110
Figure 5.5 Comparison of Point Load and Distributed Load Shear Diagrams......... 112
Figure 5.6 Comparison of Point Load and Distributed Load Moment Diagrams .... 112
Figure 5.7 Initial Parameter Variation Vertical Force Results: Mass....................... 116
Figure 5.8 Initial Parameter Variation Lateral Force Results: Mass ........................ 116
Figure 5.9 Initial Parameter Variation Vertical Force Results: Slipper Gap ............ 117
Figure 5.10 Initial Parameter Variation Lateral Force Results: Slipper Gap............ 117
Figure 5.11 Initial Parameter Variation Vertical Force Results: Vertical Rail Roughness ............................................................................................. 118
Figure 5.12 Initial Parameter Variation Lateral Force Results: Vertical Rail Roughness ............................................................................................. 118
Figure 5.13 Initial Parameter Variation Vertical Force Results: Lateral Rail Roughness ............................................................................................. 119
Figure 5.14 Initial Parameter Variation Lateral Force Results: Lateral Rail Roughness............................................................................................................... 119
Figure 5.15 Initial Parameter Variation Vertical Force Results: Vertical Natural Frequency .............................................................................................. 120
Figure 5.16 Initial Parameter Variation Lateral Force Results: Vertical Natural Frequency .............................................................................................. 120
Figure 5.17 Initial Parameter Variation Vertical Force Results: Lateral Natural Frequency .............................................................................................. 121
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Figure 5.18 Initial Parameter Variation Lateral Force Results: Lateral Natural Frequency .............................................................................................. 121
Figure 5.19 Initial Parameter Variation Vertical Force Results: Torsional Natural Frequency .............................................................................................. 122
Figure 5.20 Initial Parameter Variation Lateral Force Results: Torsional Natural Frequency .............................................................................................. 122
Figure 6.1 2212222 Vertical Best Power Form Prediction vs η Force..................... 136
Figure 6.2 1111331 Vertical Worst Power Form Prediction vs η Force .................. 136
Figure 6.3 Vertical Π Theorem Analysis.................................................................. 139
Figure 6.4 Lateral Π Theorem Analysis ................................................................... 140
Figure 6.5 2-Dimensional Interpolation Example Depiction.................................... 142
Figure 6.6 3-Dimensional Interpolation Example Depiction.................................... 143
Figure 7.1 Vertical Force Extreme Case, 1111331, Comparison ............................. 150
Figure 7.2 Vertical Force Typical Case, 2222222, Comparison............................... 151
Figure 7.3 Lateral Force Extreme Case, 1111133, Comparison............................... 153
Figure 7.4 Lateral Force Extreme Case, 2222221, Comparison............................... 153
Figure 7.5 Lateral Force Typical Case, 2222222, Comparison ................................ 154
Figure 7.6 Design Study Sled Structure.................................................................... 156
Figure 7.7 λ Slipper Beam Shear Diagram............................................................... 158
Figure 7.8 λ Slipper Beam Moment Diagram .......................................................... 159
Figure 7.9 λ Sled Body Shear Diagram.................................................................... 159
Figure 7.10 λ Sled Body Moment Diagram.............................................................. 160
Figure 7.11 SIMP Slipper Beam Shear Diagram...................................................... 165
xxiii
Figure 7.12 SIMP Slipper Beam Moment Diagram ................................................. 165
Figure 7.13 SIMP Sled Body Shear Diagram........................................................... 166
Figure 7.14 SIMP Sled Body Moment Diagram ...................................................... 166
Figure 7.15 η Slipper Beam Shear Diagram............................................................. 169
Figure 7.16 η Slipper Beam Moment Diagram ........................................................ 170
Figure 7.17 η Sled Body Shear Diagram.................................................................. 170
Figure 7.18 η Sled Body Moment Diagram ............................................................. 171
Figure 7.19 CO Test Vertical Force Comparison ..................................................... 175
Figure A.1 Rocket Motor Beam FEM ...................................................................... 194
Figure A.2 Slipper Beam Three Dimensional FEM ................................................. 195
Figure A.3 Slipper Three Dimensional FEM............................................................ 196
Figure A.4 Payload Three Dimensional FEM .......................................................... 197
1
1. INTRODUCTION
The Holloman High Speed Test Track (HHSTT) propels test items along a
steel rail using rocket sleds. This range of test items and test velocities is very broad.
Typical test items include aircraft egress systems, aerothermal test samples and
ballistic impact items. Test velocities range from static, 0 feet per second (fps), to
hypersonic in ground level air. A sled is a structure designed to interface with a
payload while being propelled by a rocket motor and its trajectory is controlled by
slippers whose inside profiles are slightly larger than the rail over which it slides.
This profile difference is shown in Figure 1.1 and allows the sled to fly through the
slipper gap should impact, inertia, and/or lift forces incline the sled to do so. Three
distinct sled configurations are used: monorail, dual rail wide gauge, and dual rail
narrow gauge. Of the three, the narrow gauge configuration has been used most
recently for high velocity impact tests and was the configuration used to set the
current land speed record, Figure 1.2.
1.1 Existing Theory
Since the inception of the HHSTT in 1950, sled designers have sought to
accurately predict structural loading experienced by the sled as it moves down the
track. The accepted design process is to hold the sled in dynamic equilibrium at
significant events during the sled test and analyze force and stress distribution
throughout the sled. Significant events have been identified from measurements and
2
Figure 1.1 Slipper Rail Interaction
Figure 1.2 Hypersonic Narrow Gauge Sled
3
represent design conditions that reduce the infinite number of possible time steps
during a test to about seven cases. This allows a designer to define load conditions
and then choose extreme design load cases to begin structural analysis. Design loads
are divided into two categories, quasi-steady-state and dynamic. Quasi-steady-state
loading is defined as loading that is well defined in magnitude and time of
application. This loading type, although time varying, may be applied at instants in
time as if essentially static with an applicable dynamic load factor to compensate for
its application time. Examples of this loading type are aerodynamic, thrusting,
inertial, and braking. Dynamic loading is defined as loading due to flexible body
interaction, slipper/rail interface, and rocket motor transients. Dynamic loading for
the current design process is applied as a static load factor, λ units of gs, through the
Center of Gravity (CG) of the free body diagram or as an acceleration vector in a
Finite Element Model (FEM). This process allows design engineers to quickly
analyze a design using static sled models. A quick process is needed since during the
course of a design cycle the analysis is typically performed several times on
converging iterations of the final sled design.
1.1.1 Lambda, λ
Dynamic load prediction on rocket sleds has a history as long as the HHSTT
itself. The earliest organized effort was accomplished by the Interstation Supersonic
Track Conference Structures Working Group, ISTRACON, in 1961 with the
publishing of the ISTRACON handbook (ISTRACON Handbook 1961). This
4
handbook comprehensively detailed track testing among the major test tracks of the
United States and recommended using ‘Rail Roughness Load Factor’ (ISTRACON
Handbook, 1961, p. 4-1-19), λ, to account for the rail roughness effects imparted to
the sled. It was defined as a function of velocity only and applied to the sled CG.
This was the first statement of the dual rail λ factor and its original intent to account
for rail roughness. In the development of λ, very basic load measurements were
taken for a wide range of speeds for different sled designs as shown in, Figure 1.3.
Typically a load sensor, force transducer or strain gage, was placed somewhere on a
sled and the reaction forces were measured and recorded during the course of a sled
test. To analyze the force data, known forces were then applied to a sled free body
diagram in static equilibrium for a given fixed time step. The forces measured at
different points on the sled were balanced and any unbalance was corrected by
applying a force at the CG to bring the forces back into equilibrium. The force
applied to the CG was divided by the sled weight and the resulting value was termed
λ. Monorail λ factor loading was first documented by Mixon (1971) where a few
measured data points were reported. The monorail λ factor was shown to be
significantly higher than dual rail λ factor. Narrow gauge λ factor was first stated by
Krupovage, Mixon & Bush (1985) and studied in the context of a damped sled test by
Hooser (1989). The listing by Krupovage et al. (1985) was a simple fit of narrow
gauge λ factor essentially halfway through that of the monorail and dual rail wide
gauge λ values as seen in Figure 1.4. This methodology was loosely confirmed
5
through one sled test by Hooser (1989). In all three sled configurations lateral values
of λ were prescribed as 0.6 of vertical λ values.
1.1.2 SIMP/SLEDYNE
The absence of known dynamic load data combined with the need for high
velocity testing prompted Mixon (1971) to undertake his work in the area of high
fidelity modeling of monorail sleds which began the next significant area of dynamic
load prediction: transient FEM modal analysis. Transient FEM modal analysis is a
numerical method where rocket sled flexibility is represented as a modal model and
excited by traversing a section of rough rail. The interaction between the sled and rail
1
10
100
0 1000 2000 3000 4000 5000 6000 7000
VELOCITY (FT/SEC)
LAM
BD
A F
AC
TOR
(G)
Background Information OnlyNot To Be Used For
Design Purposes
BTM 1 Narrow Gage Vert
BTM 1 Narrow Gage Lat
BTM 2 Narrow Gage Lat
BTM 2 Narrow Gage VertNOTS Monorail
SSIP Narrow Gage
High G Narrow Gage Vert
High G Narrow Gage Lat
Standard Rain ErosionMonorail
30" Narrow Gage
Modular Monorail
200 g Long
200 g Lat
200 g Vert
Chaparral Monorail Pusher
23" IHJ Long
High PerformanceMonorail Pusher
23" IHJ Lat
23" IHJ Vert & Twin Zap
16" Twin GILA Lat
16" Twin GILA Vert& High Performance Rain
Erosion Monorail
Chaparral
AFWL Flyer PlateDual Rail Sleds
Narrow Gage
Monorail Sleds
Figure 1.3 λ vs. Velocity with Measured Sled Data (Mixon and Hooser 2002)
6
0
5
10
15
20
25
30
0 500 1000 1500 2000 2500 3000
Velocity (ft/sec)
λ (G
)
Dual Rail
Monorail
Narrow Gauge
Figure 1.4 λ vs. Velocity
is approximated by using a coefficient of restitution formation (Abbot 1967) or
stiffness and damping formation (Mixon 1971; Greenbaum, Garner & Platus 1973;
Tischler, Venkayya & Palazotto 1981). Typically, a flexible modal model of a rocket
sled is loaded with thrust and drag as it travels along the rough rail profile (Abbot
1967; Greenbuam et al. 1973; Tischler et al. 1981). Mixon (1971) divided the
problem into two parts; the first part consists of a rigid body representation of a sled
that traverses the rough rail generating a force time history at each slipper. In the
second part, the force time histories are applied to a flexible modal model of the
rocket sled to generate force time histories at discretized sled masses. All of the
preceding methods utilized surveyed rail roughness data in only the vertical direction;
7
monorail sleds were studied by Abbot (1967), Mixon (1971), and Greenbaum et al.
(1973) and wide gauge dual rail sleds were analyzed by Mixon (1971), Greenbaum et
al. (1973), and Tischler et al. (1981). Transient FEM modal analysis was shown to
produce inertial loading that was verified with corresponding sled tests for vertical
and pitching motion. Mixon (1971) used a measurement scheme where a custom
transducer was built into the front and aft slippers of the sled. This transducer
featured three self-compensated strain gage bridges which made it possible to directly
measure the vertical and lateral force and pitching moment. The force data was used
to correlate a mathematical model in the time and frequency domains. However, the
computational and rail survey limitations that existed over 20 years ago prevented any
large scale design parameter variations and subsequent correlation. The only studies
that sought to significantly study parameter variation were those by Greenbaum et al.
(1973) and Tischler et al. (1981). The work by Greenbaum et al. (1973), a published
software package SLEDYNE, was parametrically varied to produce Sled IMpact
Parameter (SIMP) which used a series of slipper impact velocity estimates, sled mass,
and vertical stiffness to produce a SIMP factor and ultimately a maximum slipper
force. SIMP has two major drawbacks: it uses only 400 feet of vertical rail roughness
data and does not correlate well to sled test data in the lateral degree of freedom. The
parametric study of a dual rail wide gauge sled by Tischler et al. (1981) utilized
SLEDYNE and a statistical representation of rail roughness while varying slipper
beam stiffness This study did not present any correlation between peak loading and
design parameters.
8
1.1.3 Numerical Integration of Equations of Motion, NIEOM
Recently dynamic sled prediction has advanced where modeling was
improved by incorporating higher fidelity models which utilize more design
parameters and more detailed rail survey information by using Numerical Integration
of Equations of Motion (NIEOM). This method utilizes a numerical solution of the
equations of motion as generated by a commercial-off-the-shelf-package, in this case
Dynamic Analysis and Design System (DADS) (Hooser 2000, 2002b, Hooser and
Schwing 2000), to produce displacement, velocity, acceleration, forces, and moments
at each discretized mass. Rail roughness was defined in the vertical and lateral
directions using actual survey data for the length of rail the sled will traverse. Slipper
impacts were approximated using Hertzian contact theory, coefficient of restitution,
or springs and dampers. NIEOM has been successfully used to verify numerical
predictions with measured sled test data on monorail sleds (Hooser and Schwing
2000) and for narrow gauge sleds (Hooser 2002b; Minto 2004, 2002, 2000). Sleds
may be modeled as Spring Mass Damper (SMD) systems (Hooser and Schwing 2000,
Hooser 2002b, Furlow 2004), imported modal representations (Furlow 2004), or
combinations of both. DADS has been shown to match favorably to other numerical
solutions for a basic step loaded cantilever beam by Furlow (2004) and in
approximating small deflections of a vibrating nonlinear flexible body by Eskridge
(1998).
9
1.2 Limitations of Current Methods
HHSTT design engineers are currently using λ generated dynamic loads
(Krupovage, Mixon & Bush 1991) even though a significant amount of work has
been done to study the nature and parameters influencing dynamic loading. The
primary reason for this occurrence is the basic one step process associated with λ
factor as opposed to an advanced empirical method such as SIMP. Also
computational studies such as SLEDYNE rely on computing platforms and FEM
code that is no longer operational and custom computer codes have been lost or are
no longer functional. Additionally the λ method has proven to be successful in many
past sled designs. However, Mixon (1971), Mixon and Hooser (2002), and Hooser
(1989) have shown that, for certain sled configurations, λ under predicts measured
dynamic loading. This is best demonstrated in a focused look at monorail and narrow
gauge λ data versus sled test data as shown in Figures 1.5 and 1.6. The intent of λ
was to bound the highest dynamic values. In Figures 1.5 and 1.6, it can be seen that λ
significantly under predicts dynamic loading in some cases and over predicts in
others. Also in many cases the widely used ratio of lateral to vertical λ of 0.6 is not
present. The sensitivity of dynamic loading to design parameter variation was
investigated numerically by Nedimovic (2004), and sensitivity to changes in rail
roughness, slipper gap, and slipper beam stiffness were documented through a limited
design parameter study in DADS for a hypersonic narrow gauge sled. Even though
the λ method in the past has been successful, it does not account for all significant
10
design parameters nor does it produce optimized sled designs. Mixon and Hooser
(2002) attribute its success to overly conservative material safety factors and assumed
worst case load conditions that compensate for its lack of specific accuracy.
1.3 Problem Statement
Sled designers need a dynamic load prediction tool that is easily applied while
maintaining accuracy over the range of significant design parameters. The λ factor
method is extremely efficient and easily used but does not account for sled design
parameters other than velocity and mass where it has been shown experimentally
1
10
100
0 500 1000 1500 2000 2500 3000 3500 4000
VELOCITY (FT/SEC)
λ (G
)
Background Information OnlyNot To Be Used For
Design Purposes
BTM 1 Narrow Gage Vert
BTM 1 Narrow Gage Lat
BTM 2 Narrow Gage Lat
BTM 2 Narrow Gage Vert
SSIP Narrow Gage
High G Narrow Gage Vert
High G Narrow Gage Lat
30" Narrow Gage
Vertical λ
Vertical λ Lateral λ
Figure 1.5 Narrow Gauge λ Compared to Sled Tests
11
1
10
100
0 1000 2000 3000 4000 5000 6000 7000
VELOCITY (FT/SEC)
λ (G
)
Background Information OnlyNot To Be Used For
Design Purposes
NOTS Monorail
Standard Rain ErosionMonorail
Modular Monorail
200 g Lat
200 g Vert
Chaparral Monorail Pusher
High PerformanceMonorail Pusher
23" IHJ Lat
23" IHJ Vert & Twin Zap
16" Twin GILA Lat
16" Twin GILA Vert& High Performance Rain
Erosion Monorail
Chaparral
AFWL Flyer Plate
Vertical λ Lateral λ
Vertical λ
Figure 1.6 Monorail λ Compared to Sled Tests
(Mixon 1971, Hooser 1989, ISTRACON 1961) and numerically (Nedimovic 2004)
that slipper gap, sled flexibility, slipper/rail interaction, and rail roughness are also
significant and worthy of inclusion. Previous numerical and experimental attempts to
produce a dynamic load prediction tool have not thoroughly accounted for a broad
spectrum of design parameters since computational tools were not adequate or the
sheer number of experimental data points was impossible to obtain. The use of a
validated numerical model as a platform to conduct large scale parameter variation
studies solves the initial problem of obtaining a large enough sample of dynamic load
data. The next step, to produce a dynamic load prediction tool, is to correlate the
numerical data to the sled design parameters. Finally the development of a factor or
12
load to be applied at the sled CG in the manner of λ must be completed. A design
prediction tool which marries the efficiency of λ with the robustness of a detailed
wide ranging numerical study would produce an efficient and accurate tool which
would ultimately make HHSTT sled designs more optimized and reliable thus
reducing cost and increasing performance.
13
2. DYNAMIC LOAD PREDICTION OF NARROW GAUGE SLEDS
Rocket sled dynamic load prediction in its various forms from λ (ISTRACON
1961), to SLEDYNE/SIMP (Mixon 1971; Greenbaum, Garner & Platus 1973;
Tischler, Venkayya & Palazotto 1981) to NIEOM (Hooser 2000, 2002b, Hooser and
Schwing 2000) has at its core the goal to make use of known values in order to
accurately extrapolate or interpolate to unknown values of design parameters and
associated loading. The successful characterization of known dynamic loads serves
as a verification process tool for a prediction tool. The characterization criteria have
varied with each method and were heavily dependent upon the measurement
technology available at that time.
2.1 Narrow Gauge Sled Dynamic Load Prediction
Dynamic load prediction is necessary and essential for a sled designer. The
intense dynamic environment generates significant loading in comparison to
aerodynamic, thrusting and braking loads. The current method to generate a dynamic
load, which is applied statically on a sled representation in dynamic equilibrium, is
depicted in Figure 2.1. The dynamic load is applied to the sled CG and reacted by the
front and aft slippers. The generation and utilization of this dynamic load comes
from one of the three methods currently employed at the HHSTT, with λ being the
most common.
14
Figure 2.1 Application of Vertical λ to a Narrow Gauge sled
Y Vertical
Z Downtrack
X Lateral
Thrust
Drag
Wλvertical
GEW FYF
FYR
CG
15
The application of λ involves the calculation of the λ factor from the
appropriate value (monorail, narrow gauge, or dual rail) and applying the factor to the
mass of the sled. Early on in its formation, λ was applied only at the sled CG. With
wide spread use of FEM, λ has been applied as an acceleration vector to all masses in
a linear static model.
The development of SLEDYNE and SIMP (Mixon 1971; Greenbaum et al.
1973; Tischler et al. 1981) was a more involved process where initial verification was
performed by Mixon (1971). Subsequent development of SLEDYNE and SIMP
relied heavily on the initial work performed by Mixon and ultimately came full circle
with the publishing of SIMP factors (Greenbaum et al. 1973, Krupovage, Mixon &
Bush 1991) and the implementation of SLEDYNE at the HHSTT.
The application of SIMP is for preliminary design and is a two step process.
The first step involves the estimation of a slipper impact velocity based on lift-to-
weight and effective impact frequency. In the second step, the impact velocity is then
applied to a SIMP chart or equation resulting in the calculation of a maximum force
located at the slipper beam is calculated (Greenbaum et al. 1973, Krupovage et al.
1991). SLEDYNE is a more involved extension of SIMP and utilizes a modal
representation of the sled and then outputs loads at slipper beams for analysis
(Greenbaum et al. 1973, Krupovage et al. 1991). SLEDYNE considers the test
conditions for a FEM generated flexible body where SIMP is a rough estimate of the
same for a rigid body. Both methods are well correlated in the vertical direction.
NIEOM utilizes a flexible structural representation of a sled traversing a
16
rough rail. This dynamic loading calculation method is verified using an array of
measurement points located across a sled that include accelerometers, force
transducers, and strain gages. The verification of this method is detailed in Chapter 4
and can be summarized as a comparison of standard deviation in the time domain and
spectral power over a range of banded frequencies. Load prediction from this method
is accomplished by first constructing a verified model and then varying its design
parameters to represent the design in question. Maximum loading and acceleration at
critical structural components is the main output of this method and it is most useful
to the dynamic equilibrium analysis methods utilized by sled designers.
2.2 Selection of Dynamic Load Prediction Method
In comparing the currently available dynamic load prediction methods, λ,
SIMP, and NIEOM, the most easily applied is the λ factor method where the most
accurate is NIEOM. The method that is the most efficient in extrapolation and
interpolation between design parameter values is SIMP as more design parameters
have been factored into its parametric study. The NIEOM method represents an
accurate method of modeling to study, in depth, the behavior of the sled. However its
current implementation in DADS proves to be cumbersome for the design process.
Considering the strengths and weaknesses of the methods developed over the
existence of the HHSTT, a reasonable choice of a dynamic load prediction tool is one
that is as easily applied as λ and as accurate as NIEOM. The feature that makes λ not
as robust or comprehensive in its accuracy is that it is not based on the results of a
17
limited number of sled tests where design parameters have not varied greatly. The
parameter variation begun with the development of SIMP lends itself to the idea that
the required sled tests can be performed by numerical simulation using NIEOM and a
robust load prediction tool produced with the proper analysis of the design parameters
and resulting loads. The load prediction tool selected for this study is NIEOM where
its results are to be studied and manipulated into a form that can be applied as easily
as λ.
2.3 Development of Dynamic Load Prediction Method
The development of this new method entails the following tasks: 1)
Construction and validation of a DADS model using reasonable engineering
judgment, valid assumptions, and detailed structural characteristics of a representative
sled test where suitable data exists. 2) Wide scale design parameter variation to
perform many numerical simulations of sled tests. 3) Correlation of contributing
design parameters to dynamic loading outputs; and 4) Evaluation of design tool
effectiveness compared to previous methods.
2.3.1 Construction and Validation of DADS Model
This initial task is essential to the success of the method since methodology
developed in the construction and validation of the representative model will
propagate throughout the entire effort. The representative sled test for this work is
the Land Speed Record (LSR) test conducted 30 April 2003 at the HHSTT where a
18
narrow gauge forebody achieved a peak velocity of 9,465 fps. Suitable data was
recorded for this test and has been studied in depth along with very detailed
aerodynamic and structural information regarding the forebody sled. Also used in
this effort was a developmental Coast Out (CO) sled test of the same forebody where
suitable data was collected during the peak velocity of 5,343 fps and subsequent coast
out.
2.3.2 Wide Scale Design Parameter Variation
This portion of the dynamic load prediction effort involves identifying design
parameters and documenting a reasonable range for each parameter consisting of a
high, middle, and a low value. The parameter variation will be divided into two
steps. The first step is to assign all design parameters a middle value and document
the peak dynamic force at the sled CG. Then each parameter will be varied
independently to its high and low value. The change of the peak dynamic output at
the sled CG with respect to the middle value will be used to establish the influence
that each design parameter has on peak dynamic load. A suitable threshold will be
established and all parameters whose change is above the threshold will be deemed
contributing and will be retained in the next step, all others will be deemed
noncontributing and will not be considered in the next step. Also at this point any
groupings or nondimensional combinations of parameters should be identified. This
approach has two advantages. The first advantage is that the relationships that might
exist between parameters will give more insight into their influence. The second
19
advantage is that nondimensional grouping of design parameters may reduce the total
number of numerical simulations in the subsequent step of design parameter
variation.
The next step of design parameter variation is to vary all remaining
parameters through their high, middle, and, low values for all combinations. The
peak dynamic loading output associated with the sled CG and slippers will be
recorded along with design parameter values. It should be noted that from previous
work (ISTRACON 1961, Mixon 1971, Greenbaum et al. 1973, Hooser 1989, Hooser
2000, 2002, Nedimovic 2004) that velocity is a design parameter. Its variation is
inherent in every sled run as a sled begins at rest and approaches its maximum
velocity over a continuum of velocity values.
2.3.3 Correlation of Design Parameters to Dynamic Loading Outputs
This effort uses the peak dynamic loading values recorded in the previous task
to build a relationship between model inputs (design parameters) and outputs (peak
dynamic loading) at the sled CG. This relationship is the core of the dynamic load
prediction tool and will be utilized by the design engineer to predict dynamic loading
for the force/stress analysis in the sled design process. Care should be taken to ensure
that the prediction tool does not produce unreasonable values at interpolated or
extrapolated points between design parameters.
20
2.3.4 Evaluation of Design Tool Effectiveness Compared to Previous Methods
This task will document a basic comparison of a typical design where the
dynamic load will be generated by the new tool and previous methods, SIMP and λ.
The dynamic load values themselves will be compared and then the effect on overall
sled design will be noted. It is worth noting that weight, high speed sled design, is the
primary design driver. A reduction in weight allows higher velocity and more
flexible sleds which in turn produce lower loads and ultimately less stress than their
more rigid and heavier counterparts.
21
3. DEVELOPMENT OF NARROW GAUGE SLED MODEL
The narrow gauge sled model was developed in DADS to replicate the
propulsive LSR and CO forebody sleds. The methodology employed was to
construct the sled from the same information available to the design engineer using
drawings (DWG 2002E37504), Computer Aided Design (CAD) models, and
vibration test results. Since the sled in question has been tested several times and
studied in depth during its development, there was a large amount of data available to
characterize the major sled components. The ultimate goal of the DADS modeling
effort was an accurate numerical model of the sled in Figure 3.1 and served as the
starting and ending point for model development. The initial effort was to identify all
functional aspects of the sled. Several novel design concepts were employed on this
particular sled: weight and moment optimized slipper beams, lateral vibration
isolation on slipper beams, composite rocket motor case, slipper anti-rotation, lateral
and downtrack vibration isolation on the payload. Also the interior contact surface of
all slippers was configured to disallow any contact between the slipper and the
interior surface of the rail. Thus all lateral contact was on the exterior portion of the
rail. Considering all functional aspects of the sled design, several modeling
assumptions were made to ensure fidelity to sled function and to discard all
noncontributing phenomena.
22
Figure 3.1 LSR Forebody Sled
3.1 Modeling Assumptions
Initially the simulation boundaries were defined via assumptions as to what
the DADS model would and would not contain. Justification of each assumption was
tersely stated and measured against common sense. The DADS model accounts for
the items in Table 3.1 and excludes items in Table 3.2.
3.2 Design Parameter Selection
Design parameters are defined as any sled or rail characteristic that affects
sled performance. Conceivably any parameter from frequency response to sled color
could affect performance; however, in the scope of this study only those parameters
23
Table 3.1 Modeling Assumptions Ensuring Model Fidelity
Assumption Description
Elastic Slipper Rail Contact Slipper rail impact does not account for energy loss due to plastic deformation,
wear, or gouging
Flexible Body Modeling of Structural Components
Flexible body representation of rocket motor, slipper beams, slippers, and payload
Rough Rail Use survey data to vary the surface of the ideal rail profile
Rigid Rail
Rail is assumed to be rigid based on data from Baker and Turnbull (1999), Hooser (2000a), Hooser and Mixon (2000), and
Graf, K. F., Mahig, J., Wu, T. S., Barnes, R. A., Kowal, C. R. (1966)
Table 3.2 Modeling Assumptions Outside Scope of Present Study
Assumption Description
No plastic deformation at slipper rail contact Model only elastic contact
No slipper rail gouging
For all tests in LSR series rail coating and alignment technology have
prevented significant gouges (Turnbull and Minto 2003, Cinnamon 2006)
No significant heat transfer from Aerodynamic Heating or friction
Sled test does not last long enough to allow heat transfer that would affect
structural properties
No frictional wear of slippers Slipper wear is not enough to change slipper gap or structural properties of
slippers
No variations in rail stiffness The rail is uniformly rigid
24
which influence structural response and sled performance are studied as they relate
directly to the parameters that design engineers consider. Aerothermal effects are
important, but are outside the scope of this investigation. The design parameters
chosen in this study are as follows:
1) Slipper Gap, Vertical and Lateral
2) Rail Roughness, Vertical and Lateral
3) Slipper Beam Stiffness
4) Rocket Motor Stiffness
5) Component Connection Structural Characteristics
6) Component Weight
7) Structural Damping
8) Vibration Isolation
9) Component Material
10) Thrust
11) Aerodynamic Lift and Drag Forces
12) Velocity
13) Slipper Rail Contact
14) Combustion Instability
The items contained in this list represent the highest level categorization of design
parameters. For example the slipper rail contact category is made up of several
variables that include contact scheme and all of the parameters related to it.
25
3.3 Slipper Rail Impact Modeling
Slipper rail impact describes a range of high sliding velocity contacts from a
flat plate on rail, Figure 3.2, to a rotated plate on rail, Figure 3.3. The angle that the
sled can move is governed by the distance between the front and aft slipper edges and
the slipper gap. For a specialized case with the current study a 10 inch long rotated
slipper moving through a 0.125 in gap gives a pitch impact angle of about 1o. The
rotated slipper effect is worth mentioning since the impact scenario is different from
that of a flat plate with 0o angle of impact. With the rotated slipper, the leading edge
will contact first, experience some minor wear, and rotate until a 0o angle is achieved
and eventually rebound. An edge contact followed by a momentum dissipation
through slipper rotation are two additional elements that occur which affect the before
and after impact momentum with a nonzero impact angle.
The end product of slipper rail impact modeling was implemented into DADS
in the form of three contact points per slipper surface, Figures 3.4 and 3.5. This
required validation against the method used to generate the contact scheme. DADS
allows the implementation of a modified Hertzian contact scheme (Stronge 2000), or
spring and damper values, linear and nonlinear. The linear and nonlinear contact
forces are straightforward as shown in Equations 3.1 and 3.2 respectively (LMS
CADSI 2004). The Hertzian contact force is calculated as shown in Equation 3.3
(LMS CADSI 2004) with the inclusion of effective Young’s Modulus (Stronge 2000).
This range of contact phenomena was investigated from three perspectives: linear,
nonlinear and Hertzian.
26
Figure 3.2 Slipper Rail Impact Parallel Orientation
Figure 3.3 Slipper Rail Impact Maximum Angled Orientation 3o
27
Figure 3.4 Front View of DADS Slipper Rail Contact Depiction
Figure 3.5 Isometric View of DADS Slipper Rail Contact
6
5
4 3
2
1
28
Pn cVkF −= δ (3.1)
s/in) (lbt coefficien dampinglinear a is c/in)(lbt coefficien springlinear a isk
(in) deflection is
(in/s) n velocitypenetratio theis V
f
f
p
δ
where
Ppn VVckF )()( −= δδ (3.2)
s/in)(lb n velocitypenetratio offunction a as dampingnonlinear a is )c(V/in)(lb deflection offunction a as stiffnessnonlinear a is )k(
fp
fδwhere
5.1
2
2
nom
21
5.2tanh111
1733.0
11
δactn
e
pnomact
KF
VV
CORCORKK
cEK
RRc
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+−
+=
=
+=
(3.3)
)(lb deflection and K offunction a as force normal theis F/in)(lb stiffness actual theis K
/in)(lb stiffness nominal theis K
(in/s) velocity nal transitio theis VnRestitutio oft Coefficien is
)/in(lbbody contacting of Modulus sYoung' effective theis
in 84.245 is bodiesflat for 2,body contacting of Radius theis (in) 1body contacting of Radius theis
factn
0.5-fact
0.5-fnom
e
2f
1
1
CORE
RRwhere
29
3.3.1 Theoretical Formulation
The first perspective consisted of a purely theoretical standpoint to uncover
the nature of slipper rail impact. There were several sources found during the
literature review that treated in great detail the theory related to less complicated
contact such as two spheres colliding or a single sphere contacting a semi infinite
plate, Hertzian contact (Stronge 2000). A theoretical source that dealt directly with
high velocity sliding contact was not found. The literature surveyed dealt with elastic
impacts of comparatively basic systems or with complex systems that analyzed wave
propagation within the material (Zukas, J. A., Nicholas, T., Swift, H. F., Greszczuk,
L, B. 1982). Note that the latter system was discounted due to experimental work
performed by Hooser (2002a) and Hooser and Mixon (2000) and computational work
by Furlow (2006a) that showed that slipper impacts on the rail are of short duration,
typically 1 millisecond, so there is not time for the rail response to influence the
contact either initial or subsequent at sled velocities above 4,000 fps. The natural
mode of the rail in the vertical direction, 310 Hz 1st bending, and lateral direction, 199
Hz 1st bending and 490 Hz 1st torsional, give a minimum response time that cannot
structurally affect slipper rail interaction for a 130 in long sled traveling above 4,000
fps. Also shown were the formation of certain elastic waves that propagate along the
rail at given velocities (Baker and Turnbull 1999). These waves were shown to
quickly decay and not influence slipper rail interaction as it was concluded that if they
were significant the front slipper would disturb the rail and induce a response that
could destroy the following aft slipper; from observation and decades of testing this
30
scenario does not prevail. It was surmised at this point in the research effort and
noted in the literature (Baker and Turnbull 1999) that a considerable study was
warranted to truly understand slipper impacts from a theoretical standpoint. The
study would involve the broader range of slipper rail contact scenarios and not
necessarily the solid mechanics involved in the slipper interpenetrating the rail.
3.3.2 Computational Formulation
The second perspective was to approximate the macro level slipper rail impact
by studying the pre- and post-impact conditions of a slipper impacting the rail.
Several transient FEM software packages are available to perform the analysis
portion of this study. PRONTO, a FEM package developed by Sandia National Labs
(SNL) was chosen for its availability and capability to be run locally on a Linux
cluster or at the DOD High Performance Computing Center. The PRONTO models
were constructed to represent three general cases of slipper rail impact: vertical down,
lateral, and vertical up. The slipper rail model was constructed using two solid parts
separated by about 0.001 inches in the contact direction. The solid parts were
produced in I-DEAS and meshed in CUBIT, a SNL mesher. The mesh size
discretization for the slipper was 0.009 in and the rail was 0.35 in and is depicted in
Figures 3.6, 3.7, and 3.8. The tie-downs, mechanical fixtures that fasten the rail to a
concrete girder, were approximated by fixing four inch sections of the rail every 52 in
as depicted in Figure 3.9. Initially the downtrack velocity was studied at 0 fps where
several phenomena were noted. The impact of a slipper at a tie-down vs. rail mid-
31
span of the tie-downs gave different deflections and contact times but yielded similar
CORs. When the downtrack velocity was increased, the impact point became less
significant as the slipper remained in contact with the rail over tie-down and mid-span
alike. In fact at 5,000 fps, the lateral case bounding impact points show very little
difference as depicted in Figure 3.10. As a result of this a downtrack velocity of 5000
fps was used for all subsequent PRONTO analyses. The initial conditions for all
impact cases, vertical down, vertical up, and lateral, were a range of velocities of 10,
25, 50, and 100 ips in the respective impact direction with the downtrack velocity of
5000 fps. These initial conditions were intended to span the range of typical slipper
impacts as noted by Hooser and Mixon (2000). A similar analysis for all cases was
performed where a set of nodes on the slipper were identified that defined the mid-
span and ends of the slipper. The displacement, velocity, acceleration and contact
force of these node sets were monitored to describe the impact behavior of the
slipper; the gross slipper motion was derived from these data values. The contact
force was used to define the initiation of the contact as force was zero until first
contact. At the first contact point, the approach velocity and relative deflection levels
were noted as they were used in subsequent analyses. The return of the slipper
deflection to the relative deflection point was used to define where the slipper left the
rail. At this point, the rebound velocity was monitored. There was a substantial
amount of high frequency vibration present in velocity data after contact, but
typically for the period immediately after slipper rail separation, the averaged velocity
was used to define the rebound velocity of the slipper. The approach and rebound
32
velocities were used to define the Coefficient of Restitution. The time where the zero
velocity point occurred was used to ratio the force and relative displacement at this
point to determine a maximum stiffness value if the force data were significant across
the impact duration. These analysis points are depicted in a typical result for the non-
sliding 100 ips vertical up case shown in Figure 3.11 (Furlow 2006a).
The results of this study are shown in Table 3.3 and document the values of
COR. The results are averages over all of the initial conditions for each respective
impact direction. It should be noted that the COR values were simply a ratio of the
pre-impact to post-impact slipper momentum. The values compare well to
Figure 3.6 Slipper Rail PRONTO Model Front View
33
Figure 3.7 Slipper Rail PRONTO Model Side View
Figure 3.8 Slipper Rail PRONTO Model Isometric View
34
Figure 3.9 Slipper Rail PRONTO Model Boundary Conditions
-1.00E-02
-8.00E-03
-6.00E-03
-4.00E-03
-2.00E-03
0.00E+00
2.00E-03
4.00E-03
0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 6.00E-04 7.00E-04 8.00E-04 9.00E-04 1.00E-03
Time (s)
Def
lect
ion
(in)
100 ips On Tie Down
100 ips Mid Span
Figure 3.10 Lateral Impact 100 ips, 5000 fps Sliding Velocity Comparison
35
-8.00E-04
-6.00E-04
-4.00E-04
-2.00E-04
0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
1.20E-03
0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 6.00E-05 7.00E-05 8.00E-05 9.00E-05
Time (s)
Ver
tical
Dis
plac
emen
t (in
)
-8.00E+01
-6.00E+01
-4.00E+01
-2.00E+01
0.00E+00
2.00E+01
4.00E+01
6.00E+01
8.00E+01
1.00E+02
1.20E+02
Forc
e (lb
f)/ V
eloc
ity (i
ps)
Displacement
Force
Velocity
Impact
Velocity Initial
Velocity Final
Figure 3.11 Vertical Up Impact 100 ips, 0 fps Sliding Velocity
those derived from the work of Hooser and Mixon (2000) in the vertical up impacts
since momentum lost to the rail is typically 2.2 lbf s and is somewhat lower in the
vertical down as the typical momentum loss to the rail was 1.7 lbf s. The lateral
momentum loss was 2.7 lbf s which is substantially lower than the value of 32.3 lbf s
reported by Hooser and Mixon (2000), this discrepancy is most likely due to the
nature of the measured sled test. The sled test generating the experimental data was a
monorail where vertical up and down impacts are similar to a single slipper, but
lateral impacts are dissimilar in that monorails have a significant roll motion
component that adds to torsional and lateral rail deflection. Also discrepancies may
arise from assumptions made in using measured flange translational velocities to
describe a torsional rail response and too flexible of a model to represent rail motion.
36
The effective mass of both the rail and slipper were calculated and applied to the
measured velocity (rail) and SIMP derived velocity (slipper) respectively. The
important conclusion of this study was the measurement of the COR. In this study
the stiffness and damping values were adjusted since their eventual implementation
was on a rigid rail in DADS.
Table 3.3 PRONTO Slipper Impact Results
Impact Direction Coefficient of Restitution, COR
Lateral 0.24
Vertical Up 0.49
Vertical Down 0.42
3.3.3 DADS Implementation
The results of the PRONTO modeling were emulated in a DADS
representation of the test cases conducted in PRONTO. The DADS model included
assumptions to be used on the full scale sled test model. The most important among
these assumptions was the rigid rail assumption. Three different contact schemes
were studied: linear stiffness, nonlinear stiffness, and DADS implemented Hertzian as
shown in Equations 3.1, 3.2, and 3.3 respectively. The actual contact scenario is
bounded by a semi-infinite cylinder (angled slipper) contacting a semi-infinite plate
(rail) and by a finite plate (parallel slipper) contacting a semi-infinite plate (rail). In
37
contrast Hertzian contact, as implemented by DADS, represents the contact of two
spherical bodies where the contact radius corresponding to a flat surface is 84.245 in.
To represent the contact, values for stiffness, damping, COR, radius contact point,
and Young’s modulus were prescribed as necessary. The slipper was modeled as a
flexible body using modal or SMD representations. The rail was described as a rigid
body with the profile shown in Figures 3.4 and 3.5 with no rail roughness. The initial
conditions were 10, 25, 50, and 100 ips in the respective contact direction and 0 fps in
the downtrack direction. The 0 fps downtrack velocity was valid as the rail was
modeled as rigid and friction was not included in this study. The stiffness and
damping for each direction are listed in Tables 3.4, 3.5, and 3.6 for each of the
contact schemes. The measure of correlation in each case is based on a visual
inspection of PRONTO and DADS data overlays of displacement and velocity. It
should be noted that correlation of the Hertzian contact scheme, Table 3.6, was not
acceptable for any case and was thus discarded. In DADS Young’s modulus had to
be scaled to get results within a reasonable range of the PRONTO generated results
and the deflection did not correlate well. The linear and nonlinear contact schemes,
Figures 3.12 and 3.13 respectively, yielded good comparisons to the PRONTO data.
The comparisons were evaluated on two criteria: the maximum deflection and impact
duration. The COR values generated in PRONTO were used as a double check to
ensure that the DADS correlation data matched in velocity as well, and that
momentum transfer was similar to that in PRONTO. The experimental study by
Hooser and Mixon (2000) was used as a measure of effectiveness, and the DADS
38
results compared well as they were similar to the PRONTO results. It should be
noted that for nonlinear contact the force-deflection in DADS was described by a
force deflection curve from a text file. For some simple contact scenarios, the
nonlinear contact scheme produced force error and inexplicably the contact
displacement was not affected. It is surmised that this was due to interpolation errors
in DADS when calculating force and displacement from the text file, and although the
nonlinear contact scheme yielded excellent results it was not used in the complete
sled model for this reason.
3.4 Flexible Body Modeling
Flexible body representation was included in the simulation of the LSR test as
it was shown to contribute significantly in transient rocket sled models when it was
incorporated as a modal representation (Mixon 1971, Greenbaum et al. 1973) or as a
SMD representation (Hooser and Schwing 2000, Hooser 2002b, 2000). Modal
representation refers to representing a flexible rocket sled by decomposing the sled
into major structural components, applying the appropriate boundary conditions and
then solving for natural frequencies and mode shapes. Typically the solution of the
natural frequencies and mode shapes is accomplished using a finite element technique
where the model is formed by resolving mass properties at nodes and developing the
stiffness matrix based on the connecting geometry. The SMD system involves
prescribing stiffness and damping values for degrees of freedom between point
masses. The stiffnesses are formed based on the structural properties between the
39
Table 3.4 DADS Slipper Impact Results, Linear Contact Model
Impact Direction
Measured Coefficient of Restitution,
COR Stiffness (lbf/in)
Damping (lbf s/in) Correlation
Lateral 0.20 1200000 350 Good/Fair
Vertical Up 0.35 38057 60 Good
Vertical Down 0.50 5959008 600 Excellent
Table 3.5 DADS Slipper Impact Results, Nonlinear Contact Model
Impact Direction
Measured Coefficient of Restitution,
COR Stiffness (lbf/in1.5)
Damping (lbf s/in) Correlation
Lateral 0.18 40000000 400 Fair/Poor
Vertical Up 0.30 1256.5 64 Good/Fair
Vertical Down 0.50 7.39E+08 500 Excellent
Table 3.6 DADS Slipper Impact Results, Hertzian Contact Model
Impact Direction
Measured Coefficient
of Restitution,
COR
Young’s Modulus (lbf/in2)
Contact Radius
(in)
Rail Radius
(in)
Transition Velocity
(ips) Correlation
Lateral 0.24 4.0E9 0.001 84.245 1.0 Poor
Vertical Up 0.35 3.50E7 0.001 84.245 1.0 Fair
Vertical Down 0.50 5.29E9 0.001 84.245 1.0 Fair/Poor
40
-2.50E-03
-2.00E-03
-1.50E-03
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
1.50E-03
2.00E-03
0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04
Time (s)
Dis
plac
emen
t (in
)
-60
-40
-20
0
20
40
60
Velo
city
(ips
)
Dads DisplacementPronto DisplacementPRONTO VelocityDADS Velocity
Figure 3.12 Vertical Down Impact 50 ips, Linear Contact
-2.50E-03
-2.00E-03
-1.50E-03
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
1.50E-03
0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04
Time (s)
Def
lect
ion
(in)
-60
-40
-20
0
20
40
60
Vel
ocity
(ips
)
Dads dataPronto FinePRONTO VelocityDADS Velocity
Figure 3.13 Vertical Down Impact 50 ips, Nonlinear Contact
41
point masses in a manner similar to the finite element technique. The principle
difference between the two methods is that mode shapes are prescribed by their
boundary conditions regardless of their final application; however, it is possible to
group many different boundary conditions in the final application. Both methods will
be considered and analyzed in the current study and the more accurate of the two will
be selected for use in the wide scale parameter variation study.
For both methods the general implementation was to establish a set of
acceptance criteria for each major structural component in order to measure the
fidelity of the flexible body modeling scheme. The properties measured were mass
distribution, static deflection, and natural frequencies and mode shapes. The most
desirable data was that derived during laboratory testing of the actual sled
components. This data was available for natural frequencies and mode shapes for the
rocket motor body. The next most desirable source of data was from a three-
dimensional FEM representation of the component. This allowed three-dimensional
effects to be accounted for in the evaluation of static deflection and frequency data.
When a major structural component was modeled in DADS using either flexible body
representation, the DADS model characteristics were measured against the
acceptance criterion. The acceptance criterion and respective sources are listed in
Table 3.7. The most descriptive example of this method is the validation process of
the rocket motor model.
The geometrical shape of the rocket motor was produced in I-DEAS and a
typical density prescribed. The rocket motor consists of a carbon composite case with
42
Table 3.7 DADS Model Acceptance Criterion
Acceptance Criterion Major Structural
Component
Mass Static Deflection Frequency Data Slipper
Beam FEM Model FEM Model FEM Model
Rocket Motor
Laboratory Measurement Not Used Laboratory
Measurement
Payload FEM Model FEM Model FEM Model
Slipper FEM Model FEM Model FEM Model
insulating layers protecting the inside of the carbon composite from the burning
propellant. The composite lay-up was studied to determine a suitable density for the
I-DEAS model where the rocket motor casing with no propellant weight was
measured in the lab. Using the solid volume calculated in I-DEAS, a weight density
of 0.0672 lbf/in3 was prescribed. The total volume was next divided into 12 separate
volumes whose CGs each corresponded to a mass lumped at a node placed at the
corresponding CG location. Beam elements were prescribed to represent the
geometry spanning the CG locations and used area, length, shear area coefficient, and
area moments of inertia based on the cross sectional properties to approximate
flexibility. Polar moment of inertia was exactly one half of the typical value as the
carbon composite lay-up was such that this was necessary. The frequency data
criterion were used to verify a model representing a reproduction of the modal test
43
accomplished by Mixon (2002b) the results of which are featured in Table 3.8 for the
rocket motor entries. From the verified model, frequency data (Mixon 2002b),
fabrication information (Track DWG A0406003, Mixon 1999a), and consulting
Herakovich (1998) a smeared composite material value for Young’s Modulus,
11.75E6 lbf/in2, and Poisson’s ratio, 0.28, were determined along with an average
composite material layup ratio value of 0.473. It was reasoned that if frequency
response and mass properties matched laboratory test equivalents then no static
deflection validation was needed for the rocket motor. The procedure was executed
in a similar fashion for the remaining major structural components whose acceptance
criteria are listed for mass properties in Table 3.9, static deflection in Table 3.10, and
frequency data in Table 3.8. During the course of performing this analysis, care was
taken to ensure the model discretization process was adequate to represent frequency
and deflection phenomenon. Any three dimensional effects or such as translational or
rotational coupling were included in the modeling through refined substructuring or
representative stiffness. The three dimensional FEM models for the payload, slipper
beam, and slipper are shown in Appendix A. Studies were conducted on the
application of different boundary conditions to the modal model to investigate their
effect on the motion of the exported model. The findings of these side studies
showed that the boundary condition values should be based on the interfaces of the
major structural component, e.g., whether or not a connection could transmit a
moment or force. For an interface that was nonlinear it was more desirable to let the
major structural component be unconstrained in that degree of freedom so that force
44
generation would occur within DADS with the application of the nonlinear boundary
conditions.
Table 3.8 Major Structural Component Frequency Properties Major
Structural Component
Boundary Condition Frequency (Hz) and Mode Shape
Slipper Beam
Fixed at Yoke
259 SS V*
487 SS
DT*
650 C
DT*
732 1st C V*
1380 1st T*
1860 2nd C V*
2580 1st C L*
Slipper Fixed at SB
interface
708 B L*
2280 B V*
Rocket Motor
Free – Free
146 1st B V*
146 1st B L*
236 1st T DT*
381 2nd B
V*
482 1st
DT*
Payload
Fixed (FWD) –
Free (AFT)
151 1st B V*
152 1st
B L*
803 2nd
B V*
813 2nd
B L*
3500 1st
DT*
* V-Vertical; L-Lateral; DT-DownTrack; SS-See Saw; B-Bending; T-Torsional; C-Cantilever
Table 3.9 Major Structural Component Mass Properties Major Structural Component
Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
Slipper Beam 49.68 1.29 5.29 5.85
Slipper 27.06 0.770 0.803 0.521
Rocket Motor
203.15 1038.46 1033.47 31.00
Payload 145.43 194.00 194.00 7.78
45
Table 3.10 Major Structural Component Static Deflection Properties Deflection (in)/ Load Direction (lbf)
Major Structural
Component Load Point Boundary Condition X Y Z
Slipper Beam
Inside slipper
attachment
Fixed yoke
-4.78E-5 (X)
1.58E-4(Y)/ 100 X
1.58E-4 (X) 9.99E-4
(Y)/ 100 Y
4.10E-4 (Z)/ 100 Z
Slipper Lip edge Fixed at
SB interface
2.13E-3/ 100 Y
1.77E-3/ 100 Y N/A
Rocket Motor N/A N/A N/A N/A N/A
Payload N/A N/A N/A N/A N/A
3.4.1 Modal Representation
Modal representation as applied to this study was implemented in DADS by
utilizing natural frequencies and mode shapes generated in I-DEAS using beam
elements based on sled geometry. The process of generating the mode shapes in I-
DEAS and implementing them in DADS is quite complex and worthy of explanation.
Sled geometry was used to generate the length and cross sectional properties of the
major structural sled components. This information was used to produce nodes and
linear 2-node beam elements. The beam theory used to produce the stiffness
properties of the beam elements was the default method in I-DEAS (Unigraphics
Solutions Inc. 2002) and was derived from Przemieniecki’s (1968) treatment of
Timoshenko beam theory more recently documented by Timoshenko, Young, and
Weaver (1974). The Timoshenko beam equation is shown in Equation 3.4 and leads
46
to the elemental stiffness matrix seen in Appendix B. The cross sectional area, area
moments of inertia, and shear area ratios (inverse of the Timoshenko shear area
coefficient) were all computed from the prescribed element cross section. Some
investigation was made into the effect of basing the calculation of shear area
coefficient solely on geometric properties resulting in the shear area ratio being
written as the ratio of shear strain at the cross section centroid to the average shear
strain (Timoshenko et al. 1974). A study by Gruttmann and Wagner (2001)
investigated the effect of including Poisson’s ratio in the determination of shear area
coefficient and found that the effect was noticeable only for extreme values of
Poisson’s ratio coupled with thick rectangular cross sections where the height was
much less that the width. However, in the present study it was found that the shear
area coefficient calculation used by I-DEAS was more than adequate.
2
2
2
2
22
4
4
4
22
4
2
2
4
4
xq
GAEI
tq
AGIq
txyI
ty
txyE
GI
tyA
xyEI
∂∂
Ξ−
∂∂
Ξ+
=∂∂
∂−⎥
⎦
⎤⎢⎣
⎡∂∂
−∂∂
∂Ξ
−∂∂
+∂∂
ρ
ρρρρ (3.4)
force appliedan is qtcoefficien areashear Timoshenko theis
modulusshear isG area sectional cross isA
density theis axis deflected about the inertia ofmoment area theis I
modulus sYoung' is E
Ξ
ρ
where
47
After constructing a beam FEM and validating its mass distribution, static and
frequency response to either laboratory measurements or full three dimensional
FEMs, its mode shapes, natural frequencies, mass distribution, geometric and other
FEM information were exported into several text files to be processed by a command
line executable, DFBT.exe (DADS Flexible Body Translator). This executable
assembled the text files into a file that was processed using the DADS/Flex program.
Within DADS/Flex, the mode shapes and natural frequencies were orthogonalized so
that they could be run in a stable and efficient manner within DADS. The theory
behind orthogonalization is presented by LMS CADSI (1999) and seeks to ‘decouple
the modal mass matrix with respect to the modal stiffness matrix’ and is seen in
Equation 3.5. This process removes rigid body mode shapes and backs out the mass
and stiffness matrices from modal information (LMS CADSI 1999). It also may
include constraint and static attachment modes in the model so that static stiffness is
retained in the DADS model as well if the analyst requires these conditions.
( )χ
χλΨ=Ψ
=−o
mm 0MK (3.5)
matrixmation r transforeigenvecto nxm ransformedlinearly t modes new
seigenvalue of vector theis izationorthogonalfor selected modes theare
matrix stiffness modal theis matrix mass modal theis
χ
λΨo
where
m
m
ΨKM
48
The next step in the process was to prescribe a flexible body in DADS and use
the orthogonalized mode shapes defined in DADS/Flex to represent the body
flexibility and geometry. This lengthy process was verified by comparing the
structure’s mass and fundamental natural frequencies as computed in DADS to the
applicable acceptance criterion from Section 3.4. An example of this process was
demonstrated on the rocket motor.
The details of the modal model verification reveal a two part process where
first the I-DEAS model must be successfully measured against the acceptance
criterion. Secondly the information exported to DADS must be measured against the
same acceptance criterion to ensure that the exported process did alter the
information. The rocket motor model was measured in I-DEAS against mass
properties and frequency data as no lab static deflection data existed and no three
dimensional FEM was available. The other major structural components relied on
high fidelity three dimensional FEMs. The rocket motor was discretized by
prescribing lumped masses at nodes. Elements based on cross sectional properties
were prescribed. After much iteration, it was discovered that due to the composite
layup the polar moment of inertia was one-half of the typical value. The model was
modified to match the lab test scenario where mass and frequency data were produced
and are compared in Tables 3.11 and 3.12 for verification. The successful
comparison of the frequency data gives confidence to the smeared composite material
properties. At this stage the model was modified to represent a typical empty rocket
motor case. Next fixed boundary conditions were applied only to the aft slipper beam
49
interface and natural frequencies were solved for and exported to DADS. The mode
shapes were processed for import allowing the resulting comparisons to be shown in
Table 3.12 where no significant deviation is seen. This process was repeated for the
remaining major structural components except three dimensional FEMs were used as
the verification data source.
Table 3.11 Modal Model Empty Case Rocket Motor Mass Properties Comparison
Model Weight
(lbf) Ixx
(lbf s2 in) Iyy
(lbf s2 in) Izz
(lbf s2 in)
CAD Solid 203.15 1038.46 1033.47 31.00
DADS/Flex 202.68 1035.00 1030.00 30.85
Deviation % -0.23 -0.33 -0.34 -0.48
Table 3.12 Modal Model Rocket Motor Frequency Comparison for Vibration Test, Mixon (2002b)
Model Boundary Condition Frequency (Hz) and Mode Shape
Beam FEM Free - Free
146 1st B V*
146 1st B L*
236 1st T DT*
381 2nd B
V*
482 1st DT*
DADS/Flex Free – Free
141 1st B V*
151 1st B L*
241 1st T DT*
396 2nd B
V*
504 1st DT*
Deviation % -3.4 3.4 2.1 3.9 4.6
* V-Vertical; L-Lateral; DT-DownTrack; B-Bending; T-Torsional
50
3.4.2 Spring Mass Damper Representation
SMD representation was implemented by using the DADS beam element to
describe the flexibility between discretized masses. The DADS beam element uses
Euler-Bernoulli beam theory (LMS CADSI 2004) as its default to describe the
stiffness between masses in all six-degrees-of-freedom. The use of the DADS beam
element is similar to that of a finite element except that the DADS beam element is
written in terms of relative deflection where the deflection and applied load of one
beam end are relative to the other end. The values required by DADS for use in the
Beam Force element are cross sectional area, area moment of inertia, and polar
moment of inertia. Euler-Bernoulli beam theory is shown in Equation 3.6 and is valid
for beam problems where the ratio of length over effective depth is greater than 10. It
was shown for suddenly applied loads (Furlow 2004) that the Euler-Bernoulli beam
theory was acceptable even though individual beam components did not meet this
requirement as long as the entire structure met the length to depth ratio. The
implementation of Euler-Bernoulli beam theory by DADS is shown in Appendix A
and details a reduced 6 by 6 matrix reflecting DADS’ use of a relative force and
deflection implementation. For cross sections that varied between discretized masses
a factoring method was used to modify the applied cross section properties. The
factoring method accounts for the construction of multiple DADS Beam Force
elements between masses each representing a portion of the entire structure between
the masses. The typical treatment of springs in series was applied for each DOF that
a particular stiffness represented. The goal was to produce equivalent values of area,
51
area moment of inertia, and polar moment of inertia so that one DADS Beam Force
element could be prescribed between masses to represent the cross section variations.
The general equation applied to determine each equivalent value is shown in Equation
3.7. This method was used on DADS SMD modeling of the rocket motor, payload,
and slipper. This process was verified by determining the structure’s mass,
fundamental frequency, and static deflection and comparing them to the validation
data. An example of this process is described for the major structural component
rocket motor.
The rocket motor was discretized into 12 masses whose total values are shown
in Table 3.13 and is identical to the acceptance criterion as the DADS model was
discretized directly from the acceptance criterion. The structural stiffness was
defined using DADS Beam Force elements. The mass properties of the entire model
were processed and compared to the verification data. The frequency response was
measured by altering the rocket motor case to reflect the test documented by Mixon
(2002b) where torsional exciters, 40 lbf each, were placed in both ends of the empty
rocket motor case and a force exciter, 51.00 lbf, was attached to the forward torsional
exciter to apply transient loading to the rocket motor case. The entire test setup was
suspended on bungee cables to represent a free-free condition. The same situation
was recreated in DADS and the resulting natural frequencies compared to the
acceptance criterion in Table 3.14 where a maximum deviation of 11.3% occurs. The
application of this modeling process to the remaining major structural components
yielded favorable comparisons and are provided in Appendix B. The simplicity of the
52
SMD modeling method is that it is not affected by boundary conditions as is the case
with modal methods. If verification is achieved for the SMD method in mass
properties, static deflection, and frequency for a particular set of boundary conditions,
then it should be acceptable for all applied boundary conditions and any error should
arise primarily from the accuracy of the method used for its structural flexibility.
),(2
2
4
4
txqtyA
xyEI =
∂∂
+∂∂ ρ (3.6)
force appliedan is qarea sectional cross isA
density theis axis deflected about the inertiaplanar theis I
modulus sYoung' is E
ρ
where
)...(...)...()...()...(
112121
21
−+++++
++
+++=
niinniiiniii
totalniiiEquivalent LLL
Lψψψψψψψψψ
ψψψψψ (3.7)
sections beam ofnumber total theisn propertieselement beam theof axis theis j
sections beam ofnumber theis iend beam toend beam fromproperty entire the torefers total
subscriptby designatedlength is L
I A, :propertysection crosselement force beam is jjψwhere
53
Table 3.13 SMD Model Mass Properties
Model Weight
(lbf) Ixx
(lbf s in) Iyy
(lbf s in) Izz
(lbf s in)
CAD Solid 203.15 1038.46 1033.47 31.00
DADS SMD 203.16 1038.46 1033.47 31.00
Deviation % 0.00 0.00 0.00 0.00
Table 3.14 SMD Model Frequency Properties
Modal Boundary Condition Frequency (Hz) and Mode Shape
Lab Data/ Beam FEM
Free - Free
146 1st B V*
146 1st B L*
236 1st T DT*
381 2nd B
V*
482 1st
DT*
DADS SMD
Free – Free
144 1st B V*
147 1st B L*
241 1st T DT*
424 2nd B
V*
493 1st
DT* Deviation
% -1.4 0.7 2.1 11.3 2.3
* V-Vertical; L-Lateral; DT-DownTrack; B-Bending; T-Torsional 3.5 Sled Model Formation
The formation of a sled model consisted of assembling the major structural
components of the sled using appropriate connections to join them. The assembly
was accomplished in DADS by approximating the as-built function of the sled
observed from test conditions. Since the major structural components represented
directly determinable entities, there were several connections that were not directly
determinable due to 3 dimensional structural effects and nonlinearities, chiefly
54
structural and boundary conditions. The areas that were analyzed and included in the
model as connectors for the major structural components are shown in Figure 3.14.
External loading, aerodynamic and thrust, were applied as described by Minto and
Turnbull (2003) to duplicate the velocity profiles shown in Figure 3.15 (LSR) from
4.73 to 6.03 seconds (4,647- 9,465 fps) and Figure 3.16 (CO) from 1.46 to 2.20
seconds (5,327 –5,004 fps). Rocket motor mass and mass properties were varied
linearly within DADS to account for propellant mass loss during thrusting.
3.5.1 Three Dimensional Structural Effects
The three dimensional structural effects were seen primarily in the lateral/roll
motion of the payload as the deflection of its cross section changed significantly due
to a very thin wall thickness and very stiff forward attachment point located at the
bottom of the cross section. These effects were implemented in DADS by tuning its
attachment stiffness so that its DADS response matched the corresponding three
dimensional FEM response in I-DEAS. The other area of significant three
dimensional structural effects was the aft end of the payload where an plate vibration
effect was observed in the three dimensional FEM and was adjusted in the same
manner in the DADS model by using tuned attachment stiffnesses.
3.5.2 Nonlinear Boundary Conditions
Nonlinear boundary conditions (BC) contribute significantly to the response of the
model and were therefore carefully considered. The areas and types of nonlinear BCs
55
Figure 3.14 Sled Structural Connection Depiction
Slipper Attachment
Aft Payload Attachment
Slipper Beam Attachment
Fwd Payload Attachment
Side View
Front View
Slipper Beam
Slipper
Payload
Rocket Motor
Instrumentation Pallet
56
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 1 2 3 4 5 6
Time (s)
Vel
ocity
(fps
)
Velocity vs Time
Figure 3.15 LSR Velocity Profile
0
1000
2000
3000
4000
5000
6000
0 0.5 1 1.5 2
Time (s)
Vel
ocity
(fps
)
Velocity vs Time
Figure 3.16 CO Velocity Profile
57
are listed in Table 3.15. The lateral vibration isolation at the connection of the slipper
to the slipper beam represented a preloaded compression-only pair of differing
stiffness nonlinear springs. This configuration is depicted in Figure 3.17 and shows
an 80A durometer polyurethane pad of uncompressed length 0.56 in between the
slipper beam and inner portion of the slipper with a 90A durometer polyurethane pad
of uncompressed length 0.50 in. The implementation involved evaluation of the test
time connecting bolt preload, in this case 8.5 kips, and determining the
precompression length for both springs using Equation 3.8. It should be noted that
the isolation materials 80A and 90A durometer polyurethane both have high damping
values and nonlinear hardening stiffnesses as reported by Mixon (2002a) and shown
in Figure 3.18. The stiffness in this connection; however, was linearized about the
preload value of 8.5 kips as the load at this connection remains near the preload value
during the test as reported by Turnbull and Minto (2003). The damping values were
calculated to be 30 lbf s/in for both pads.
A similar nonlinear boundary condition occurred at the aft payload attachment
where a vibration isolation mechanism utilized an 80A durometer polyurethane pad
as shown in Figure 3.19. On one side of the mechanism the pad was 1.00 in thick
with an area of 7.07 in2 and on the other side was a 0.50 in thick washer with an area
of 0.76 in2. This resulted in an effective one sided compression-only spring. Initially
preload was applied to the thick pad but as the rocket motor fired motor expansion
removed the preload and consequently none was modeled. The stiffness of the thick
pad was set to 37,475 lbf/in by applying the appropriate area and length to Mixon’s
58
Table 3.15 Nonlinear Boundary Condition Summary
Nonlinear Boundary Condition Location Type
Lateral vibration isolation at slipper/slipper beam interface
Nonlinear stiffness, unidirectional application
Lateral and axial vibration isolation between payload and rocket motor
Nonlinear stiffness, unidirectional application
Payload moment reduction mechanism
Gap in aft payload vertical support creates a nonlinear support that actuates only after gap is traversed in the positive or negative direction
Slipper gapGap between slipper and rail that allows contact once gap has been traversed
Figure 3.17 Lateral Slipper Vibration Isolation
90A Durometer Polyurethane 80A Durometer
Polyurethane
59
Free80A 80A
compressed80A
Free90A 90A
compressed90A
5.8
5.8
δδ
δδ
−=
−=
kkips
kkips
(3.8)
inlb
k
inlb
k
where
f
f
147,561
45,559
90A
80A
=
=
0
5000
10000
15000
20000
25000
30000
35000
40000
0 0.05 0.1 0.15 0.2 0.25 0.3
Deflection (in)
Forc
e (lb
f)
80A Durometer 0.5625in thickness 7.4 in^2 area90A Durometer 0.5in thickness 13.0 in^2 area8.5kips Preload
Figure 3.18 Nonlinear Polyurethane Force Deflection Test Data (Mixon 2002a)
(2002a) stress/strain curve for 80A durometer polyurethane.
The next area of nonlinearity occurred in the aft vertical payload attachment.
The fabrication of this connection indicated a diameter of 0.80 in on the aft payload
attachment bracket connected to the payload and a diameter of 0.76 in on aft payload
60
slider vertically connected to the rocket motor. This joint is designed to be a pivot
about the lateral axis connected by a 0.75 in diameter bolt. This gives a vertical
clearance of 0.028 in when the diameter of the bolt and clearance holes of the aft
payload attachment bracket and aft payload slider are considered. This connection
results in a nonlinear spring with characteristics shown in Figure 3.20 for a linear
stiffness of 1,102,961 lbf/in used in Equation 3.9. In effect there is no vertical force
unless payload aft moves 0.06 in up or down from neutral. This concept is supported
by analyzing the aft vertical force during the LSR test (Turnbull and Minto 2003).
inlb
961,102,1 f==L
AEk (3.9)
in 1.37 n,compressioin matierial ofLength Linlb 30E6 Modulus, sYoung' E
in 1.2 n,compressioin Area A
2f
2
=
=
=
where
There was an attempt to utilize 80A durometer polyurethane vibration
isolation in the aft payload lateral attachment but after studying both the force and
acceleration test data it is apparent the as-designed vibration isolation system failed to
perform as desired. The aft payload connecting mass weighs approximately 6-8 lbf
and if not connected to the rocket motor would be loaded laterally only by lateral
acceleration. The discrepancy in reporting the weight of this item is due to the
addition of several undocumented items at run time which included accelerometers,
61
Figure 3.19 Aft Payload Downtrack and Lateral Vibration Isolation Mechanism
-40000
-30000
-20000
-10000
0
10000
20000
30000
-0.1 -0.05 0 0.05 0.1Deflection (in)
Forc
e (lb
f)
Figure 3.20 Gap Induced Nonlinear Stiffness at Payload Aft Vertical Attachment
Aft Payload Rocket Motor Attachment
Aft Payload Attachment Slider
Aft Payload Attachment Bracket
80A Durometer Polyurethane
Payload
Lateral Direction
Downtrack Direction
62
strain gauges, and wiring; the original item weighed 5.7 lbf. The typical local
acceleration peak value in this area over the test duration is 35-50 Gs and results in a
force of 210-400 lbf when applied to the aft payload connecting mass. This value is
near to the corresponding force measurement which was typically 200-700 lbf. As a
result the aft end of the payload was allowed to move freely through the aft payload
connection gap.
An instrumentation pallet was installed on this sled to record, digitize, and
telemeter data during the sled test. The instrumentation pallet consisted of electronic
components hard mounted to two 0.25 in thick aluminum plates welded together to
form a cruciform cross section inside of an aluminum cylinder as shown in Figure
3.21. The pallet was encased in polyurethane foam of varying density including 13,
15, and 35 lbf/in3. Since the pallet is much stiffer than the isolating foam it was
modeled as a rigid point mass supported by translational springs on either end. The
foam was idealized as a viscoelastic material where its stiffness and damping
characteristics were determined from the lower curve of dynamic stress strain test
data (General Plastics Manufacturing Company 2000). A precompression value of
0.25 in was used for the vertical and lateral directions and a precompression value of
1.72 in was used for the downtrack direction. The stiffness and damping values are
provided in Table 3.16. The damping values are characteristic of 13 lbf/in3 foam and
are derived from replicating the dynamic stress strain testing in DADS. The
instrumentation mass properties listed in Table 3.17 are inclusive of the foam as
detailed.
63
Figure 3.21 Instrumentation Pallet Depiction
Table 3.16 Instrumentation Pallet Foam Vibration Isolation Characteristics
Direction Stiffness (lbf/in) Damping (lbf s/in)
Vertical 1980 0.9
Lateral 1980 0.9
Downtrack 300 0.9
The most influential nonlinear boundary condition of the entire model is the
slipper rail interface. The slipper and sled are constrained to the earth by the rail
where slippers are of the same profile as the rail but over cut by an amount known as
the slipper gap. This relationship is depicted in Figure 3.22. The boundary condition
itself results in a quasi one-sided contact that allows the sled to respond in a free-free
Battery Box
Cruciform
End Cap
Foam
64
Table 3.17 Instrumentation Pallet Rigid Mass Mass Properties
Component Weight
(lbf) Ixx
(lbf s2 in) Iyy
(lbf s2 in) Izz
(lbf s2 in)
Pallet 25.30 1.80 1.78 0.30
Foam 9.77 2.51 2.51 0.40
Composite 35.07 4.31 4.30 0.70
Figure 3.22 Slipper Rail Interface Showing Slipper Gap
65
mode when flying through the slipper gap and respond in a one-sided manner when in
contact with the rail. To compound the matter further there are many combinations of
this boundary condition as there is coupling due to the angled surfaces of the sides
and bottom of the rail. This allows force propagation in both the vertical and lateral
directions during lateral and vertical-up impacts. The actual contact between the two
was discussed in detail in the preceding section and in short consists of force
generation using stiffness and damping values to approximate the impact. This
approximation was implemented in DADS using the Point-Seg-Rail contact element
where each element consists of a single point on a slipper compliantly contacting a
rigid rail. The rail profile depicted in Figure 3.4 was used to describe the rail cross
section. Rail roughness, variations of the rail surfaces along the downtrack direction,
was introduced using survey data processed for implementation into DADS. The
survey data were acquired with a laser interferometric surveying system as part of the
continuous rail alignment program carried on at the HHSTT. Rail downtrack
measurements are accounted for using a Track Station (TS) terminology where TS
0.0 is the southernmost end of the HHSTT. The survey data was taken every 26 in
and describes the deviation of the contact surface over which it is applied. The
contact numbering scheme in Figure 3.4 shows two contact surfaces on the rail top (3
and 4), one contact surface on each of the lateral surfaces (2 and 5), and one contact
surface on each of the rail head bottom surfaces (1 and 6). The two distinct surfaces
on the rail top (3 and 4) were to account for fluctuations of rail head surface. The rail
head bottom surfaces (1 and 6) used survey data from the corresponding rail head top
66
surface ( 3 and 4) as survey data from the rail head bottom is not measurable with
current survey methods. The rail roughness and associated standard deviations are
depicted in Figures 3.23, 3.24, 3.25, 3.26, and 3.27. The presence of three rail
roughness profiles for B-rail opposed to two profiles for C-rail represents the
available survey data at the time of this work.
3.5.3 External Loading, Aerodynamic and Thrusting
External loading was divided into lift, drag, and thrust. All three were applied
to their respective directional components where the magnitude versus time was input
into the model. For the LSR test the drag, Figure 3.28, lift, Figure 3.29, and thrust,
Figure 3.30, were applied. For the CO test the drag, Figure 3.31, and lift, Figure 3.32,
were applied.
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000
TS (feet)
Rou
ghne
ss (i
n)
Standard Deviation: 0.0116 in
Figure 3.23 C-Rail East Vertical Roughness, DADS Contact Points 1, 3, 4, 6
67
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000
TS (feet)
Rou
ghne
ss (i
n)
Standard Deviation: 0.0163 in
Figure 3.24 C-Rail East Lateral Roughness, DADS Contact Point 2
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000
TS (feet)
Rou
ghne
ss (i
n)
Standard Deviation: 0.0132 in
Figure 3.25 B-Rail West Lateral Roughness, DADS Contact Point 5
68
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000
TS (feet)
Rou
ghne
ss (i
n)Standard Deviation: 0.0118 in
Figure 3.26 B-Rail East Vertical Roughness, DADS Contact Points 4 and 6
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000
TS (feet)
Rou
ghne
ss (i
n)
Standard Deviation: 0.0123 in
Figure 3.27 B-Rail West Vertical Roughness, DADS Contact Points 1 and 3
69
0
3000
6000
9000
12000
15000
4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1
Time (s)
Dra
g (lb
f)
Aft Slipper BeamForward Slipper BeamPayloadVertical Wedge
Figure 3.28 LSR Component Drag
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
7000
4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1
Time (s)
Lift
(lbf)
Aft Slipper BeamForward Slipper BeamPayload
Figure 3.29 LSR Component Lift
70
0
50000
100000
150000
200000
250000
4.7 4.9 5.1 5.3 5.5 5.7 5.9 6.1 6.3
Time (s)
Thru
st (l
bf)
Figure 3.30 LSR Rocket Motor Thrust
0
3000
6000
9000
12000
15000
1 1.2 1.4 1.6 1.8 2 2.2
Time (s)
Dra
g (lb
f)
Aft Slipper BeamForward Slipper BeamPayloadVertical Wedge
Figure 3.31 CO Component Drag
71
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
7000
1 1.2 1.4 1.6 1.8 2 2.2
Time (s)
Lift
(lbf)
Aft Slipper BeamForward Slipper BeamPayload
Figure 3.32 CO Component Lift
3.5.4 Rocket Motor Mass Variation
Rocket motor propellant mass was varied linearly within DADS for the LSR
simulation during the thrust phase. DADS is a transient solver using numerical
integration at each time step to solve the equations of motion, a zero mass property
value would cause a matrix singularity. As a result of this, transient mass properties
were at full value at the start of the simulation and linearly varied to a negligible
value at the end of the simulation. The values used for the transient mass properties
are produced from discretizing the propellant mass properties and are shown in Table
3.18. The propellant mass is non structural and has no effect on stiffness, but its
effect on natural frequency of the rocket motor case is taken into account by varying
the mass and inertia value.
72
Table 3.18 Propellant Mass Properties Component Inertias (lbf s2 in) CG Location (in) Component
Name Mass (lbf
s2/in) Ixx Iyy Izz X Y Z
motor_f4 0.1098 2.04 2.04 3.58 0 0 122.80
motor_f3 0.2235 5.75 5.75 7.29 0 0 111.17
motor_f2a 0.1828 4.14 4.14 5.96 0 0 101.51
motor_f2 0.2062 5.02 5.02 6.72 0 0 93.76
motor_f1 0.2680 8.01 8.01 8.74 0 0 81.87
Motor 0.2865 9.11 9.11 9.34 0 0 68.22
motor_a1 0.2865 9.11 9.11 9.34 0 0 54.58
motor_a2 0.2865 9.11 9.11 9.34 0 0 40.93
motor_a3 0.2865 9.11 9.11 9.34 0 0 27.29
motor_a4 0.2408 6.57 6.57 7.85 0 0 13.64
Composite Total 2.9032 2643.09 2643.09 77.50 0 0 65.70
73
4. EVALUATION OF NUMERICAL MODEL AND OUTPUT CHARACTERISTIC STUDY
The narrow gauge sled model developed in DADS was studied in two areas:
model validity and output characteristics. This was done to assess model quality
against measured data sets and to give insight into model behavior with regard to
input parameters to help determine initial model sensitivities and design drivers. The
initial data set consisted of eleven channels of force, moment, and acceleration data
from the LSR test (Turnbull and Minto 2003) where its collection points are
schematically represented in Figure 4.1. The second data set consisted of sixteen
channels of force, moment, and acceleration from a 5,343 fps velocity CO
developmental test (Leadingham and Schauer 2001) where its collection points are
schematically represented in Figure 4.2. The LSR test represented a maximum
velocity case for test data and the CO test represented a lower velocity test where the
sled was allowed to coast out while data were collected with the motor not burning.
It should be noted that significant motor combustion instability effects were present
while the motor was firing and arise from high amplitude internal pressure
fluctuations occurring near 180 Hz. These effects are not modeled.
4.1 Evaluation of Numerical Model
The numerical model developed for DADS correlation required a
comprehensive validation. The data from the LSR and CO tests was considered to be
74
WL 0.00
#115 Payload Aft Attachment Vertical Force (SGB) #138 Payload Aft Attachment Lateral Force (SGB) ~STA 85.5 (Aft Hinge Axis STA 86.56, BL 0.00, WL 25.51)
All sensors are based upon strain gage bridge installations. IB = Instrumented Bolt (Commercial) SGB = Strain Gage Bridge Circuit (Locally developed) PA = Piezoresistive Accelerometer (Commercial)
STA 0.00
Payload Front Attachment STA 50.0 STA 84.32
#101 Front Slipper Beam Rocket Motor Attachment Vertical Force (SGB) BL 0.00, WL 3.75 #103 Front Slipper Right Lateral Force (IB) ~BL 13.1, WL 3.75
WL 16.07
Aft Slipper Beam CL STA 170.58 Front Slipper Beam CL STA 49.83
Pyld Aft Accelerations: STA BL WL #123 Lateral (PA) 85.16 1.00 (lt) 27.82 #124 Vertical (PA) 85.16 1.60 (lt) 28.42
Pyld Fwd Accelerations: STA BL WL #139 Axial (PA) 52.52 0.00 28.34 #140 Lateral (PA) 53.12 0.60 (rt) 28.34 #141 Vertical (PA) 53.12 0.00 28.94
CL = Centerline (lt) = Left (rt) = Right Dimensions in inches
#112 Payload Pitch Bending Moment (SGB) #113 Payload Yaw Bending Moment (SGB) STA 39.93, BL 0.00, WL29.95
WL 29.95
Figure 4.1 LSR Data Channel Schematic
s
Forward Motor ~ STA 47.0, WL 16.07#134 Vertical Acceleration (PA) #136 Axial Acceleration (PA)
Forward Slipper Beam CL Sta. 49.9, WL 3.75 #101 Motor Attachment Vertical Force (SGB) BL 0.0#102 Motor Attachment Roll Moment (SGB) BL 0.0#104 Left Lateral Force (IB) BL ~13.1 (lt)#108 Right Roll Bending Moment (SGB) BL 5.08 (rt) #109 Left Roll Bending Moment (SGB) BL 5.08 (lt)
Aft Slipper Beam CL Sta. 170.8 #105 Motor Attachment Vertical Force (SGB)#106 Right Lateral Force (SGB)#107 Left Lateral Force (SGB)
Payload Fwd Data: STA BL WL#112 Pitch Plane Bending Moment (SGB) 40.0 0.0 29.95#114 Attachment Vertical Force (IB) 50.0 0.0 29.95#113 Yaw Plane Bending Moment (SGB) 40.0 0.0 29.95#121 Vertical Acceleration (PA) 20.6 0.0 29.95#115 Payload Aft Attachment Pin Vertical Force (SGB)
#122 Payload Aft Axial Acceleration (PA) ~STA 86.0 (Hinge Axis BL 0.00 WL 25.51) Station 0
WL 29.95
WL 16.065
WL 0.00
IB=Instrumented BoltSGB=Strain Gage BridgePA=Piezoresistive Accelerometer
CL=centerline(lt)=Left(rt)=RightDimensions in inches
s
Forward Motor ~ STA 47.0, WL 16.07#134 Vertical Acceleration (PA) #136 Axial Acceleration (PA)
Forward Slipper Beam CL Sta. 49.9, WL 3.75 #101 Motor Attachment Vertical Force (SGB) BL 0.0#102 Motor Attachment Roll Moment (SGB) BL 0.0#104 Left Lateral Force (IB) BL ~13.1 (lt)#108 Right Roll Bending Moment (SGB) BL 5.08 (rt) #109 Left Roll Bending Moment (SGB) BL 5.08 (lt)
Aft Slipper Beam CL Sta. 170.8 #105 Motor Attachment Vertical Force (SGB)#106 Right Lateral Force (SGB)#107 Left Lateral Force (SGB)
Payload Fwd Data: STA BL WL#112 Pitch Plane Bending Moment (SGB) 40.0 0.0 29.95#114 Attachment Vertical Force (IB) 50.0 0.0 29.95#113 Yaw Plane Bending Moment (SGB) 40.0 0.0 29.95#121 Vertical Acceleration (PA) 20.6 0.0 29.95#115 Payload Aft Attachment Pin Vertical Force (SGB)
#122 Payload Aft Axial Acceleration (PA) ~STA 86.0 (Hinge Axis BL 0.00 WL 25.51) Station 0
WL 29.95
WL 16.065
WL 0.00
IB=Instrumented BoltSGB=Strain Gage BridgePA=Piezoresistive Accelerometer
CL=centerline(lt)=Left(rt)=RightDimensions in inches
Forward Motor ~ STA 47.0, WL 16.07#134 Vertical Acceleration (PA) #136 Axial Acceleration (PA)
Forward Slipper Beam CL Sta. 49.9, WL 3.75 #101 Motor Attachment Vertical Force (SGB) BL 0.0#102 Motor Attachment Roll Moment (SGB) BL 0.0#104 Left Lateral Force (IB) BL ~13.1 (lt)#108 Right Roll Bending Moment (SGB) BL 5.08 (rt) #109 Left Roll Bending Moment (SGB) BL 5.08 (lt)
Aft Slipper Beam CL Sta. 170.8 #105 Motor Attachment Vertical Force (SGB)#106 Right Lateral Force (SGB)#107 Left Lateral Force (SGB)
Payload Fwd Data: STA BL WL#112 Pitch Plane Bending Moment (SGB) 40.0 0.0 29.95#114 Attachment Vertical Force (IB) 50.0 0.0 29.95#113 Yaw Plane Bending Moment (SGB) 40.0 0.0 29.95#121 Vertical Acceleration (PA) 20.6 0.0 29.95#115 Payload Aft Attachment Pin Vertical Force (SGB)
#122 Payload Aft Axial Acceleration (PA) ~STA 86.0 (Hinge Axis BL 0.00 WL 25.51) Station 0
WL 29.95
WL 16.065
WL 0.00
IB=Instrumented BoltSGB=Strain Gage BridgePA=Piezoresistive Accelerometer
CL=centerline(lt)=Left(rt)=RightDimensions in inches Figure 4.2 CO Data Channel Schematic
75
mildly stationary as most other high speed rocket sled data (Mixon 1971). The
method used to compare the model data to the test data was to form statistical
comparisons of each data channel in the time and frequency domains after applying a
3rd order digital bandpass filter, 10Hz to 4500Hz, in MATLAB. The time step for the
DADS simulations was 1E-4 seconds to prevent aliasing and allow filtering up to
5,000 Hz giving an effective sampling rate of 10,000 Hz. Associated LSR data
processing MATLAB scripts are located in Appendix C; the CO data processing
scripts follow the exact algorithm but differ only to account for different channel
names. The nature of the data also dictated that direct overlays of test and DADS
data were not productive since any small shift in frequency or time step would
generate significant deviations that were not reflective of the data comparison. Also
the nature of rocket sled testing is that no two sled tests are the same since the sled
impacts the rail in a random manner. Thus the vibration levels are similar for
duplicate sleds but never observed to be identical. As a result the time and frequency
domain comparisons were conducted in a velocity or frequency window for each
respective domain. A weighted Root Mean Square (RMS) deviation Figure of Merit
(FOM) system was used to measure the correlation of the DADS data to the test data.
A FOM of 100% indicated complete correlation for that particular measurement. All
individual FOMs were averaged with equal weighting to form a single composite
FOM for the model in question for both the time and frequency domains.
76
4.1.1 Time Domain Comparison
The data comparison in the time domain consisted of parameterizing velocity
over time and evaluating data over fixed velocity windows, 250 fps wide, to allow
statistical properties to be extracted (Furlow 2006b). The reasoning behind
windowing time domain data is twofold: first velocity windowing produces locally
stationary data (Mixon 1971) allowing meaningful statistical evaluations, and second
sled test vibration data is strongly velocity dependent (ISTRACON 1961, Hooser and
Schwing 2000, and Mixon 1971). The standard deviation over the velocity window
was then calculated using the default MATLAB algorithm. This process was
accomplished for both test and DADS generated datasets. Linear curve fits for the
test and DADS standard deviation data were calculated. The individual percentage
deviation values of the DADS data from the test data curve fit were equally weighted
by an amount whose summation equaled unity and the RMS of the weighted
deviations was calculated and termed the RMS deviation for the particular data
channel. A FOM was calculated as one hundred percent minus the RMS deviation
and was used as a measure of correlation between the DADS and test data. This
process was applied to all data channels annotated in Figures 4.1 and 4.2 and FOMs
were produced for a DADS model and were then averaged into a single FOM to
measure the correlation of that particular model. An example of the standard
deviation (STD) vs. velocity overlay is shown in Figure 4.3 along with a sample of
the calculations used to produce the data in Table 4.1.
77
STANDARD DEVIATION 101(Fwd SB Vertical Force)
y = 0.0017x - 5.5667
y = -4E-05x + 4.4626
-3
-2
-1
0
1
2
3
4
5
6
7
4900 4950 5000 5050 5100 5150 5200 5250 5300
Velocity (fps)
Stan
dard
Dev
iatio
n (k
ips)
Test STDDADS STDResidual DADS Fit - TestLinear (DADS STD)Linear (Test STD)
Figure 4.3 Time Domain Overlay, FOM 84.07%
Table 4.1 Time Domain Comparison Calculation Test Data DADS Data % %
Velocity Stdev Test Fit Velocity Stdev DADS Fit Weighted Weighted(fps) (G) (G) (fps) (G) (G) Residual wt factors Deviation Deviation^2
5249.37 4.67 4.98 5252.05 2.63 2.73 -2.35 0.14 6.75 45.535211.10 4.27 4.89 5201.77 3.50 2.73 -1.39 0.14 4.07 16.525153.03 5.75 4.74 5151.44 2.26 2.72 -2.48 0.14 7.48 55.915097.75 4.84 4.60 5101.50 2.24 2.72 -2.36 0.14 7.32 53.615052.75 5.04 4.49 5051.43 2.48 2.71 -2.01 0.14 6.40 40.935003.84 3.64 4.36 5001.51 3.18 2.70 -1.18 0.14 3.87 14.974951.85 4.10 4.23 4951.55 2.72 2.70 -1.52 0.14 5.12 26.24
0.53 1 5.86 15.93σ-res Sum of Abs % DeRms % Dev
weightfactors
4.1.2 Frequency Domain Comparison
The data comparison in the frequency domain consisted of computing a Power
Spectral Density (PSD) of the data channel and then computing the RMS spectral
78
power within discrete frequency bands (Furlow 2006b). The use of discrete
frequency bands allowed small frequency shifts to exist but not significantly affect
the comparison and also to better characterize the response of the model or test
hardware in the frequency domain. The computation of the RMS power consisted of
numerically integrating the PSD over the discrete frequency band using a simple
Romberg integration algorithm and taking the square root of the integral. The
percentage deviation of the DADS-generated data from the test data was weighted by
an amount equal to the normalized distribution of the test data RMS spectral power
from 0 to 1000 Hz. This weighted distribution allows an efficient comparison of the
DADS-generated data to the test data. The deviation at each discrete frequency band
was computed into a single RMS deviation value which was then subtracted from one
hundred to produce the FOM for the data channel in question. For each of the data
channels the frequency FOM was averaged into a single FOM to represent the
measure of correlation for the DADS model to the sled test. An example of the PSD
overlay of DADS and test data is shown in Figure 4.4 and the calculations used to
produce a FOM for the overlay is shown in Table 4.2. Discrepancies in total power
values indicate one of two things: 1) an under representation of the system frequency
response due to discretizing a continuous system, or 2) a shift in frequency response
in the model where response is greater elsewhere than at the particular measurement
location.
79
PSD 101 (Fwd SB Vertical Force)
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1 10 100 1000Frequency (Hz)
PSD
(kip
s^2/
Hz)
Test Data PSDDADS PSD
Figure 4.4 PSD Overlay, FOM 70.00%
4.1.3 Correlation Criteria
A FOM threshold was established for individual and composite FOM to
define correlation. A FOM of 100% denoted complete correlation and is only
achievable under artificial conditions since sled tests are nondeterministic.
Thresholds for frequency and time domain composite and individual FOM values
were established where greater than 80% for time domain FOM and greater than 70%
for frequency domain FOM in the composite category denoted acceptable levels
while specifying that no individual frequency domain FOM could be less than 50%
and no individual time domain FOM could be below 70%. The additional constraint
in the time domain correlation data comes from the fact that the ultimate goal of this
80
Table 4.2 RMS Power Calculation Test Data DADS Data %
Frequency RMS Power % TTL RMS Power % TTL Weight Weighted(Hz) (G rms) RMS PWR (G rms) RMS PWR Factors Deviation9.77 0.13 0.03 0.13 0.05 0.031 -0.09
24.41 0.16 0.04 0.08 0.03 0.038 -1.8948.83 0.63 0.15 1.29 0.46 0.147 15.4997.66 1.79 0.42 0.74 0.26 0.419 -24.50
151.37 0.25 0.06 0.26 0.09 0.058 0.43200.20 0.21 0.05 0.18 0.06 0.049 -0.74249.02 0.16 0.04 0.06 0.02 0.038 -2.42297.85 0.17 0.04 0.02 0.01 0.039 -3.42351.56 0.19 0.05 0.01 0.00 0.045 -4.22400.39 0.13 0.03 0.01 0.00 0.030 -2.74454.10 0.08 0.02 0.01 0.00 0.018 -1.49498.05 0.03 0.01 0.00 0.00 0.008 -0.79546.88 0.03 0.01 0.00 0.00 0.008 -0.75600.59 0.09 0.02 0.00 0.00 0.021 -2.14654.30 0.07 0.02 0.00 0.00 0.016 -1.64698.24 0.04 0.01 0.00 0.00 0.009 -0.90751.95 0.04 0.01 0.00 0.00 0.010 -1.01800.78 0.02 0.00 0.00 0.00 0.004 -0.42854.49 0.02 0.01 0.00 0.00 0.005 -0.53898.44 0.01 0.00 0.00 0.00 0.002 -0.21952.15 0.01 0.00 0.00 0.00 0.002 -0.15
1000.98 0.00 0.00 0.00 0.00 0.001 -0.094.26 2.80 30.00
Total Power Total Power Rms % Dev
study is to produce predictions for peak dynamic load and thus matching locally
stationary peak values is more important than frequency characterization although the
two are related. The only external sled forces that are not directly determinable are
slipper rail impacts so considerable attention was paid to the FOMs associated with
measurements near the slipper rail interface: channels 101 and 103 in Figure 4.1 and
channels 101, 102, 104, 105, 106, 107, 108, and 109 in Figure 4.2.
81
4.2 Output Characteristic Study
The DADS model described in Section 3.5 was constructed to emulate the
LSR and CO tests and their FOMs were monitored for correlation. Also the output of
the DADS models were monitored during the correlation effort to gain insight into
model behavior and an initial understanding of design parameters that govern this
behavior. This study consisted of monitoring the FOMs produced in the correlation
to assess sensitivity to design parameter variation. Model refinement was employed
to ensure that the model ultimately converged to an acceptable value of FOMs for
each data channel monitored. Correlated models were produced, and general output
parameter sensitivities were noted during model refinement and flexible body
formulation.
4.2.1 Modal Model Characteristics
Modal model outputs were prescribed to correspond as closely as possible to
the test data locations shown in Figures 4.1 and 4.2. Nodes were used to track
displacement, velocity, and acceleration data; force and moment data were taken at
DADS-prescribed major structural component connection elements. Nodes are FEM
point masses and were discretized in I-DEAS where other masses were represented in
DADS as a point mass. The source of DADS data outputs are detailed in Tables 4.3
and 4.4 and show the particular method and measurement type. The DADS element
TSDA refers to a translational spring. The DADS element Bracket refers to an
analyst prescribed rigid connection of two masses. The DADS element Bushing
82
Table 4.3 LSR Modal Model Output Measurement
Channel Number
Measurement Type
DADS Element Type DADS Model Element Name
101 Vertical Force DADS TSDA TSDA_MB_F
103 Lateral Force DADS TSDA TSDA_FRONT_BELL1b
112 Pitching Moment DADS Bracket BRACKET_PL_SRR_FRONT
113 Yawing Moment DADS Bracket BRACKET_PL_SRR_FRONT
115 Vertical Force DADS Bushing, Pair BUSH_LAT_SIM_ATCH_L/R
123 Lateral Acceleration Node PAYLOAD_FLEX_FLEX:node
59
124 Vertical Acceleration Node PAYLOAD_FLEX_FLEX:node
60
138 Lateral Force DADS TSDA, Pair TSDA_LAT_SIM_ATCH_R/L
139 Downtrack Acceleration Node PAYLOAD_FLEX_FLEX:node
62
140 Lateral Acceleration Node PAYLOAD_FLEX_FLEX:node
63
141 Vertical Acceleration Node PAYLOAD_FLEX_FLEX:node
64
refers to a flexible connection of two masses whose stiffnesses emulate a bushing.
All output measurements were sensitive to changes in the stiffness parameters
of the connecting elements and slipper impact parameters. The LSR output channels
139, 140, and 141 and CO output channel 122 were the least sensitive to the
parameter changes since they are physically remote from the slipper rail interaction
and the sled loading is highly attenuated along this load path. High damping at the
83
Table 4.4 CO Modal Model Output Measurement
Channel Number Measurement Type
DADS Element Type DADS Model Element Name
101 Vertical Force DADS TSDA TSDA_MB_F
102 Roll Moment DADS Bracket BRACKET_MB_F
104 Lateral Force DADS TSDA TSDA_FRONT_BELL2b
105 Vertical Force DADS TSDA TSDA_MB_A
106 Lateral Force DADS TSDA TSDA_REAR_BELL1b
107 Lateral Force DADS TSDA TSDA_REAR_BELL2b
108 Roll Moment DADS Bracket BRACKET_MB_F
109 Roll Moment DADS Bracket BRACKET_MB_F
112 Pitching Moment DADS Bracket BRACKET_PL_SRR_FRONT
113 Yawing Moment DADS Bushing, Pair BRACKET_PL_SRR_FRONT
114 Vertical Force DADS TSDA TSDA_FRONT_PL_ATCH
115 Vertical Force DADS Bracket BRACKET_PL_SRR_AFT
121 Vertical Acceleration Node PAYLOAD_FLEX_FLEX:no
de23
122 Axial Acceleration Node PAYLOAD_FLEX_FLEX:node1
134 Vertical Acceleration Node SRR_FLEX:node29
136 Axial Acceleration Node SRR_FLEX:node29
84
slipper-rail interface and vibration isolation locations tended to inaccurately reduce
acceleration levels and increase loading at the interfaces. The stiffness of the
connection at measurement 115 was factored from its original expected value of
1,102,961 lbf/in to 55,148 lbf/in. It is noted that the actual condition of this joint was
that a gap of unknown magnitude existed. Also the exact nature of this contact is not
known during the LSR and CO tests, and thus some factoring is acceptable. Output
sensitivity indicated that modal modeling was highly dependent upon mode shapes
retained for sled modeling. DADS/Flex allows the analyst to choose which mode
shapes to include for orthogonalization and these choices were made on the basis of
retaining the lower fundamental frequencies in displacements of interest and ensuring
that the DADS/Flex orthogonalization algorithm produced orthogonal mode shapes.
There were instances with all modal representations where using all available mode
shapes did not produce an orthogonal set and higher order mode shapes were rejected
until the algorithm was successful. The degree to which this phenomena impacts
accurate modal representations depends upon on the number of mode shapes retained
to produce an orthogonalized set. The methodology followed herein is to include the
maximum number of mode shapes while assuring the lower frequency mode shapes
were retained. It should be noted that DADS is not able to successfully process an
unorthogonalized set of modes shapes for flexile body analyses.
Principal strengths in using modal representation of flexible bodies is the ease
of modeling in CAD packages, and specifically in I-DEAS, the formation of beam
models includes the use of Timoshenko beam theory. The shear deflection effect, in
85
addition to the inclusion of higher order vibration effects via rotary inertia, is
automatically included in this scenario. DADS simulation times were up to 50% less
using modal representation than with the SMD representation.
Weaknesses include the dependency of the modal response on fixed boundary
conditions which for linear systems is not an issue. For this sled with several
instances of nonlinearity that are present in the interconnection of the major structural
components, it is a problem that cannot be completely solved by substructuring. This
forces the analyst to modify the parameters of connecting structures or major
structural component boundary conditions to force a particular response.
4.2.2 SMD Model Characteristics
SMD model outputs were prescribed to coincide with test data locations
shown in Figures 4.1 and 4.2. Masses were used to produce displacement, velocity,
and acceleration data where force and moment data were taken at DADS prescribed
major structural component connection elements. The source of DADS data outputs
are detailed in Tables 4.5 and 4.6 and show the particular method and measurement
type.
Principal strengths of modeling with the SMD scheme is the ability to
construct a flexible body irrespective of boundary conditions. The resulting model
will respond to nonlinear inputs or boundary conditions based upon its mass
discretization and stiffness formulation. The flexibility representation chosen for this
study is Euler-Bernoulli theory which is the default within DADS; any other theory
86
Table 4.5 LSR SMD Model Output Measurement
Channel Number
Measurement Type
DADS Element Type
DADS Model Element Name
101 Vertical Force DADS Bushing BUSHING.F_SB
103 Lateral Force DADS TSDA TSDA_FRONT_BELL1b
112 Pitching Moment DADS Beam Force Element PL_5_PL_4
113 Yawing Moment DADS Beam Force Element PL_5_PL_4
115 Vertical Force DADS Beam Force Element PL_1_PL_AFT
123 Lateral Acceleration Mass SIM_ATCH_AFT_2
124 Vertical Acceleration Mass SIM_ATCH_AFT_2
138 Lateral Force DADS Beam Force Element PL_1_PL_AFT
139 Downtrack Acceleration Mass PL_3
140 Lateral Acceleration Mass PL_3
141 Vertical Acceleration Mass PL_3
would have to be entered via matrices in text files. Earlier work (Furlow 2004)
indicated some isolated instances of DADS solver instability and overall difficulty in
managing text file input stiffness matrices. The main concern with using Euler-
Bernoulli theory with relatively short length beams is that shear deflection may be
significant and that the resulting deflection error may propagate. It was shown for a
87
Table 4.6 CO SMD Model Output Measurement
Channel Number Measurement Type DADS Element Type
DADS Model Element Name
101 Vertical Force DADS Bushing BUSHING.F_SB
102 Roll Moment DADS Bracket BRACKET_MB_F
104 Lateral Force DADS TSDA TSDA_FRONT_BELL2b
105 Vertical Force DADS TSDA BUSHING.A_SB
106 Lateral Force DADS TSDA TSDA_REAR_BELL1b
107 Lateral Force DADS TSDA TSDA_REAR_BELL2b
108 Roll Moment DADS Beam Element BEAM_F_SB_R_2
109 Roll Moment DADS Beam Element BEAM_F_SB_L_2
112 Pitching Moment DADS Beam Element PL_5_PL_4
113 Yawing Moment DADS Beam Element PL_5_PL_4
114 Vertical Force DADS Bushing BUSHING.PL_M_FOR
115 Vertical Force DADS Beam Element PL_1_PL_AFT
121 Vertical Acceleration Mass PL_CONE
122 Axial Acceleration Mass SIM_ATCH_AFT_2
134 Vertical Acceleration Mass M_FOR
136 Axial Acceleration Mass M_FOR
88
collection of short Euler-Bernoulli beams (Furlow 2004) that it was possible to obtain
deflections for the composite beam that were not affected by shear deflection effects.
4.3 Flexible Body Representation Selection
To proceed with the next phase of the study, wide scale parameter variation, a
reduced complexity model was to be constructed requiring the selection of a flexible
body representation. For the LSR test data set the best case FOM Summary is
presented in Table 4.7 for the modal representation and in Table 4.8 for the SMD
representation. It was very difficult to achieve a FOM Summary that met the
correlation criteria for the LSR test data set. This is partly due to unknown nonlinear
boundary conditions, unknown structural connections and from combustion
instability of the rocket motor. The combustion instability represented not only a
significant fluctuation in thrust but also a noticeable expansion and contraction of the
rocket motor cross section affecting data outputs at the payload attachment points.
The influence on the rocket motor cross sections is difficult to measure
experimentally and thus difficult to characterize and include in a numerical model.
The CO test data comparisons were more favorable as data were acquired during the
decelerating non-thrusting portion of the sled test that was free of combustion
instability. Thus a more direct comparison was made. The best case FOM summary
is shown for the modal representation in Table 4.9 and SMD representation is shown
in Table 4.10. The SMD flexible body formulation is the better choice for flexible
body representation as it exhibited the best overall FOM especially in the
89
measurement locations near the slippers and slipper beams and thus more accurately
describes dynamic loading from the slipper-rail interface. The SMD flexible body
formulation also displays more fidelity as it allows a more detailed description of
structural response as boundary conditions, mode shape selection, and
orthogonalization are not required for prescribing flexible body representation. The
best case FOM summary is listed in Table 4.10 and meets the minimum criteria
except for 113 which is away from the slippers and slipper beams. It should be
noted for less complex bodies with linear connections the modal representation may
be more desirable as its element formulation is superb and its solution time is
generally lower than the SMD representation.
Table 4.7 LSR Modal FOM Summary FOM
Channel Description Frequency Time101 Forward Slipper Beam Vertical Force 58.81 75.27103 Forward Slipper Beam Lateral Force 69.07 79.05112 Payload Pitch Bending Moment 63.23 84.12113 Payload Yaw Bending Moment 64.68 87.29115 Payload Aft Attachment Vertical Force 41.55 86.39123 Payload Aft Lateral Acceleration 81.68 90.31124 Payload Aft Vertical Acceleration 83.68 87.61138 Payload Aft Attachment Lateral Force 83.12 90.33139 Payload Forward Axial Acceleration 84.32 90.42140 Payload Forward Lateral Acceleration 76.25 93.77141 Payload Forward Vertical Acceleration 76.91 89.39
Composite FOM 71.21 86.72
90
Table 4.8 LSR SMD FOM Summary FOM
Channel Description Frequency Time101 Forward Slipper Beam Vertical Force 81.15 86.30103 Forward Slipper Beam Lateral Force 61.19 55.82112 Payload Pitch Bending Moment 33.61 86.47113 Payload Yaw Bending Moment 64.23 90.84115 Payload Aft Attachment Vertical Force 53.72 84.42123 Payload Aft Lateral Acceleration 47.04 86.35124 Payload Aft Vertical Acceleration 42.21 92.51138 Payload Aft Attachment Lateral Force 64.42 85.64139 Payload Forward Axial Acceleration 72.36 88.75140 Payload Forward Lateral Acceleration 57.60 93.77141 Payload Forward Vertical Acceleration 83.38 92.81
Composite FOM 60.08 85.79
Table 4.9 CO Modal FOM Summary FOM
Channel Description Frequency Time101 Forward Slipper Beam Vertical Force 63.85 82.63102 Forward Slipper Beam Roll Moment 69.86 93.78104 Forward Slipper Beam Left Lateral Force 83.51 91.49105 Aft Slipper Beam Vertical Force 70.64 86.04106 Aft Slipper Beam Right Lateral Force 84.54 87.95107 Aft Slipper Beam Left Lateral Force 81.36 91.34108 Forward Slipper Beam Right Roll Moment 78.90 88.65109 Forward Slipper Beam Left Roll Moment 76.73 91.38112 Payload Pitch Bending Moment 72.47 69.39113 Payload Yaw Bending Moment 37.07 62.50114 Payload Forward Vertical Force 69.14 70.74115 Payload Aft Attachment Vertical Force 51.98 85.48121 Payload Forward Vertical Acceleration 77.71 88.63122 Payload Aft Axial Acceleration 81.52 86.24134 Motor Forward Vertical Acceleration 76.40 80.63136 Motor Forward Axial Acceleration 61.43 74.18
Composite FOM 71.07 83.19
91
Table 4.10 CO SMD FOM Summary FOM
Channel Description Frequency Time101 Forward Slipper Beam Vertical Force 70.00 84.07102 Forward Slipper Beam Roll Moment 71.21 83.33104 Forward Slipper Beam Left Lateral Force 78.33 95.72105 Aft Slipper Beam Vertical Force 72.91 87.43106 Aft Slipper Beam Right Lateral Force 86.03 90.04107 Aft Slipper Beam Left Lateral Force 80.68 92.52108 Forward Slipper Beam Right Roll Moment 69.61 91.76109 Forward Slipper Beam Left Roll Moment 76.06 92.64112 Payload Pitch Bending Moment 66.25 84.34113 Payload Yaw Bending Moment 43.05 73.41114 Payload Forward Vertical Force 69.37 74.64115 Payload Aft Attachment Vertical Force 71.82 96.11121 Payload Forward Vertical Acceleration 79.54 95.05122 Payload Aft Axial Acceleration 71.61 95.52134 Motor Forward Vertical Acceleration 77.77 76.27136 Motor Forward Axial Acceleration 52.79 92.03
Composite FOM 71.06 87.80
92
5. EVALUATION OF LARGE SCALE DESIGN PARAMETER VARIATION
The parameter variation process was used to determine the sensitivity of the
sled model to different design inputs. After the DADS modeling process was
successfully correlated a reduced complexity model was produced with the correlated
modeling process. The model itself is similar to the third stage pusher sled used to
propel the LSR forebody to its maximum velocity. After constructing the reduced
complexity model its design parameters were varied and the corresponding solutions
were extracted. The design parameters were varied to encompass a range expected
during the normal course of track testing. Possible design parameter values included
a typical, high, and low to describe the range expected. Parameter variation was
accomplished in increasingly detailed steps indicative of the refinement needed to
ensure description of the dynamic loading outputs observed. For all parameter
variations, error tracking of the processed results was carried out to ensure that during
the execution of hundreds of simulations that excessive deflection and out of
tolerance downtrack velocity did not occur.
5.1 Construction of Reduced Complexity Parameter Variation Model
The numerical model developed for correlation in Chapter 4 served as a base
from which the reduced complexity parameter variation model was constructed.
Directly retained were the slippers along with their associated contact values which
were generated in Chapter 3 to describe slipper rail impact. The slipper beam
93
composition was made more uniform and the attachment was done through its neutral
axis to help prevent extraneous deflection coupling in the vertical and lateral
directions. Differences included a fixed mass for the sled body and no payload. The
reduced complexity produced a sled configuration more applicable to general sled
testing to ease the computational burden of the numerical solutions associated with
wide scale parameter variation. Aerodynamic loading was omitted to allow a peak
load that is purely dependent upon dynamic effects of the sled interacting with the rail
and resulting frequency responses.
The thrust applied to the sled produced a constant acceleration required to
propel the sled from 0 to 10,000 fps. The thrust is 105,275.4 lbf and is intended to
propel a 842.3 lbf sled to 10,000 fps in 2.49 seconds at 125 Gs. The resulting reduced
complexity model is depicted in Figure 5.1.
5.1.1 Sled Body
The sled body was modeled as a 136 in long cylindrical tube discretized into
12 equal evenly spaced masses having the mass properties shown in Table 5.1 for the
typical mass values. The discretization was accomplished such that all masses had
the same values representing an equal 11.33 in of the sled body length. The total
mass and associated properties are the same as a final stage rocket motor used on the
LSR and CO tests that is half way through its thrusting cycle. The thrust was applied
at the forward most mass, m_for. The flexibility between the masses was prescribed
with DADS Beam Force Elements using the constant property cross section detailed
94
Figure 5.1 Reduced Complexity Model Depiction
in Table 5.2 and applied to associated elements in the model with a structural
damping ratio of 5.0% of critical. The flexibility was prescribed from the typical
cross section of the final stage rocket motor used on the LSR and CO tests which
represented 80% of the total cross sectional area as calculated along the length of the
body. It should be noted that the relationship of the polar moment of inertia to the
area moments of inertia for the rocket motor is maintained.
Slipper
Slipper Beam
Body
Rail
95
Table 5.1 Sled Body Mass Distribution Location From Centerline and
Body Aft End (in) Mass Properties about CG
(lbf s2 in)
Name Mass
(lbf s2/in) X Y Z Ixx Iyy Izz
m_for 0.138 0.00 0.00 130.33 14.617 14.409 2.347
m_f4 0.138 0.00 0.00 119.00 14.617 14.409 2.347
m_f3 0.138 0.00 0.00 107.67 14.617 14.409 2.347
m_f2a 0.138 0.00 0.00 96.33 14.617 14.409 2.347
m_f2 0.138 0.00 0.00 85.00 14.617 14.409 2.347
m_f1 0.138 0.00 0.00 73.67 14.617 14.409 2.347
m 0.138 0.00 0.00 62.33 14.617 14.409 2.347
m_a1 0.138 0.00 0.00 51.00 14.617 14.409 2.347
m_a2 0.138 0.00 0.00 39.67 14.617 14.409 2.347
m_a3 0.138 0.00 0.00 28.33 14.617 14.409 2.347
m_a4 0.138 0.00 0.00 17.00 14.617 14.409 2.347
m_aft 0.138 0.00 0.00 5.67 14.617 14.409 2.347
96
Table 5.2 Sled Body Flexibility
Name Length
(in) Area (in2)
Iy (in4)
Iz (in4)
J (in4)
Young’s Modulus (lbf/in2)
Poisson’s Ratio
beam_for_f4
beam_f4_f3
beam_f3_f2a
beam_f2a_f2
beam_f2_f1
beam_f1_mtr
beam_mtr_a1
beam_a1_a2
beam_a2_a3
beam_a3_a4
beam_a4_aft
11.33 13.398 377.743
377.743
377.743 11.75E6 0.28
5.1.2 Slipper Beam
Both forward and aft slipper beams were modeled as 3.5 in by 3.5 in 0.375 in
thick annular box beams 36 in long. The beams were discretized into 13 equal
portions each having a length of 2.308 in as shown in Table 5.3. The flexibility
between the discretized masses was represented using DADS Beam force elements
97
applying the constant cross sectional properties shown in Table 5.4. The slipper
beams were modeled as Vascomax 300, a high nickel maraging steel, with a Young’s
modulus of 27.5E6 lbf/in2 and a Poisson’s ratio of 0.32. The flexibility was
implemented in DADS using the DADS Beam Force Element and applying the
corresponding values with a structural damping value of 3.0% of critical. Both the
forward and aft slipper beams were the same with regard to mass and flexibility
distribution.
5.1.3 Slippers
The slippers were retained directly from the correlation modeling effort for
the SMD LSR model (846th Test Squadron 2003) as they represented a successfully
employed real world standard that will be used in future narrow gauge testing. The
slipper was discretized into eight masses connected using DADS Beam Force
Elements as depicted in Figure 5.2 and listed in Table 5.5. Equivalent beam sections,
as detailed in Chapter 3, were employed in the slipper flexible body representation
and account for cross sectional changes from mass to mass. The beam section values
are shown in Table 5.6 and represent the implementation of Vascomax 300, a high
nickel maraging steel, with a Young’s modulus of 27.5E6 lbf/in2 and a Poisson’s ratio
of 0.32.
5.2 Design Parameter Formulation and Associated Values
The all inclusive list of design parameters from Chapter 3 is repeated below
98
Table 5.3 Slipper Beam Mass Distribution Location From Centerline and
Body Centerline (in) Mass Properties about CG
(lbf s2 in)
Name Mass
(lbf s2/in) X Y Z Ixx Iyy Izz
sb_l_6 0.00793 28.846 0.000 0.000 0.026 0.017 0.017
sb_l_5 0.00793 26.538 0.000 0.000 0.026 0.017 0.017
sb_l_4 0.00793 24.231 0.000 0.000 0.026 0.017 0.017
sb_l_3 0.00793 21.923 0.000 0.000 0.026 0.017 0.017
sb_l_2 0.00793 19.615 0.000 0.000 0.026 0.017 0.017
sb_l_1 0.00793 17.308 0.000 0.000 0.026 0.017 0.017
sb_c 0.00793 15.000 0.000 0.000 0.026 0.017 0.017
sb_r_1 0.00793 12.692 0.000 0.000 0.026 0.017 0.017
sb_r_2 0.00793 10.385 0.000 0.000 0.026 0.017 0.017
sb_r_3 0.00793 8.077 0.000 0.000 0.026 0.017 0.017
sb_r_4 0.00793 5.769 0.000 0.000 0.026 0.017 0.017
sb_r_5 0.00793 3.462 0.000 0.000 0.026 0.017 0.017
sb_r_6 0.00793 1.154 0.000 0.000 0.026 0.017 0.017
99
Table 5.4 Slipper Beam Flexibility
Name Length
(in) Area (in2)
Iy (in4)
Iz (in4)
J (in4)
Young’s Modulus (lbf/in2)
Poisson’s Ratio
beam_sb_l_6
beam_sb_l_5
beam_sb_l_4
beam_sb_l_3
beam_sb_l_2
beam_sb_l_1
beam_sb_c
beam_sb_r_1
beam_sb_r_2
beam_sb_r_3
beam_sb_r_4
beam_sb_r_5
beam_sb_r_6
2.307 4.688 7.739 7.739 15.479 30.00E6 0.32
100
Figure 5.2 DADS Slipper Depiction of Front Right Slipper
for ease of reference and is intended to show a comparison between it and the list of
reduced/combined design parameters in Table 5.7.
1) Slipper Gap, Vertical and Lateral
2) Rail Roughness, Vertical and Lateral
slip_low_in_frt_r
slip_mid_h_in_frt_r
slip_mid_lo_in_frt_r
slip low out frt r
slip_mid_l_out_frt_r
slip_mid_h_out_frt_r
slip_frt_rt_con
slip_cent_frt_r
101
3) Slipper Beam Stiffness
4) Rocket Motor Stiffness
5) Component Connection Structural Characteristics
6) Component Weight
7) Structural Damping
8) Vibration Isolation
9) Component Material
10) Thrust
11) Aerodynamic Lift and Drag Forces
12) Velocity
13) Slipper Rail Contact
14) Combustion Instability
For design parameters such as slipper gap and rail roughness there are values
specified by HHSTT operating procedures (846th Test Squadron 2006) that designate
a precise value range. The minimum and maximum values from this range were used
in Table 5.7. The formation of total sled free-free natural frequencies accounts for
component material, material distribution, and component stiffness for each major
structural component. Lateral slipper vibration isolation is accounted for in the
formation of the lateral natural frequency because it is considered a spring in series
with the axial stiffness of the slipper beam. It should be noted that complexity
reduction in this area included modeling the vibration isolation of equal linear type
material on both sides of the slipper; thus eliminating the need for a highly complex
and nonlinear representation of this joint as a set of nonlinear preloaded springs.
102
Table 5.5 Slipper Mass Distribution
Location From Centerline and Body Centerline (in)
Mass Properties about CG
(lbf s2 in)
Name
Mass (lbf
s2/in) X Y Z Ixx Iyy Izz
slip_cent 0.008 -0.070 -5.857 0.000 0.067 0.068 0.002
slip_con 0.004 0.590 -4.012 0.000 0.033 0.033 0.0004
slip_low_in 0.005 -4.323 -4.041 0.000 0.042 0.042 0.001
slip_mid_h_in 0.009 -3.641 -5.868 0.000 0.073 0.075 0.003
slip_mid_h_out 0.034 -1.809 -2.271 0.000 0.330 0.291 0.167
slip_low_out 0.010 -1.625 0.000 0.000 0.020 0.020 0.020
slip_mid_lo_in 0.005 0.708 -5.139 0.000 0.0389 0.038 0.001
slip_mid_l_out 0.006 -4.435 -5.107 0.000 0.048 0.048 0.001
5.2.1 Sled Mass
Sled weight refers directly to the sled mass and associated mass properties.
The high and low values are representative of composite sled values where the body
is varied from mass properties representing unexpended (high), half-expended
(typical), and expended (low) LSR final stage rocket motor mass. Consideration was
given to the fact that mass distribution was important to the performance of the sled
and was not varied by just simply scaling the mass moments of inertia. The
103
Table 5.6 Slipper Flexibility
Name Length
(in) Area (in2)
Iy (in4)
Iz (in4)
J (in4) Mass 1 Mass 2
beam_s_cent_in 2.22 5.00 41.77 41.67 0.18 slip_cent slip_mid
_h_in
beam_s_cent_out 2.70 5.00 41.77 41.67 0.18 slip_mid_
h_out slip_cent
beam_s_mid_out 1.07 6.88 57.60 57.60 0.27 slip_mid_l
_out slip_mid_h_out
beam_s_low_out 0.79 7.26 60.84 60.51 0.31 slip_mid_l
_out slip_low_
out
beam_s_mid_in 1.13 5.19 43.37 43.25 0.11 slip_mid_
h_in slip_mid_lo_in
beam_s_low_in 0.78 7.00 58.67 58.38 0.28 slip_low_i
n slip_mid_lo_in
variations of mass and mass moments of inertia were somewhat counterintuitive
because the discretized mass inertia values were tuned so that the composite body
values matched the LSR rocket motor equivalent. The sled mass high, typical, and
low values are shown in Table 5.8.
5.2.2 Slipper Gap
Slipper gap definition is: ‘Total slipper gap is defined as the distance from one
slipper bearing surface to the rail surface with the opposite slipper bearing surface in
contact with the rail. Both vertical and horizontal gap shall be checked.’ (846th Test
Squadron 2006). Currently the slipper gap range is from 0.125 +/- 0.015 in and
represents the typical, high, and, low values respectively to be used in the parameter
variation study.
104
Table 5.7 Reduced Design Parameter Listing
Values
Name Designation Low Typical High
Sled Mass (lbf) DP1 383.3 842.0 1,301
Slipper Gap (in) DP2 0.110 0.125 0.140
STD Rail Roughness, Vertical (in) DP3 0.0083 0.0119 0.0155
STD Rail Roughness, Lateral (in) DP4 0.0103 0.0147 0.0191
ω Sled V* (Hz) DP5 41.0 79.9 200
ω Sled L* (Hz) DP6 44.0 84.5 168
ω Sled TOR* (Hz) DP7 45.0 85.9 161
Velocity (fps) (windowed Δv=250
fps) DP8 0.0 5,000 10,000
* V–Vertical; L–Lateral; TOR–Torsional
Table 5.8 Reduced Complexity Sled Mass Properties
CG (in) Mass Properties about CG
(lbf s2 in)
Case Weight
(lbf)
Mass (lbf s2 /in) X Y Z Ixx Iyy Izz
Low 383.37 0.9931 0 -7.57 68.55 1967.97 1984.82 123.92
Typical 842.20 2.1816 0 -3.45 68.25 4708.14 4692.67 221.12
High 1301.02 3.3701 0 -2.23 68.16 6445.43 6422.10 268.26
105
5.2.3 Rail Roughness
Rail roughness was applied in the same manner prescribed in Chapter 3 with
the values used in correlation modeling representing typical values as surveyed from
TS 44,845 to 49,926 on B and C rails. The high and low values were represented by
+/- 30% deviation of the typical values. This variation will be applied through simple
scaling of the corresponding rail roughness within the DADS model. The
corresponding standard deviation values for the rail roughness profiles and
corresponding contact points are shown in Table 5.9 and Figure 3.4 and represent
standard deviation values as seen in Figures 3.23, 3.24, 3.25, 3.26, 3.27.
5.2.4 Sled Natural Frequency
The sled natural frequencies listed in Table 5.7 refer to modal analysis
Table 5.9 Rail Roughness Standard Deviation Values
Case Standard Deviation
(in) Direction DADS Contact
Points
High 0.0155
Typical 0.0119
Low 0.0083
Vertical 1,3,4,6
High 0.0191
Typical 0.0147
Low 0.0103
Lateral 2,5
106
solutions of the sled with free-free boundary conditions. The natural frequencies are
the fundamental, 1st, in their respective directions. This is allows a design engineer to
produce a basic FEM of an early stage sled design and quickly determine the natural
frequencies for entry into the dynamic load prediction tool. The boundary conditions
are prescribed with two intentions: First, to create an easily applied standard so that
few discrepancies can exist when the dynamic load prediction tool is applied.
Second, to account for the fact that the product of the tool, force, is applied to the sled
CG where the dynamic loading is assumed to come from the slipper/rail interaction
and the load path is from the slipper to the sled CG. This designation of boundary
conditions and resulting natural frequencies may appear somewhat artificial because
during a test the sled encounters a combination of free-free and fixed translational one
sided boundary conditions. However, the intent is to establish a standard by which all
sleds may be easily measured and a well defined stiffness and mass distribution for
the entire system can be accurately established. The corresponding reduced
complexity parameter variation model will respond to any boundary condition
combination encountered in the DADS simulation accomplishing the goal of
establishing a common, easily measured input parameter.
5.2.5 Velocity
The velocity profile desired for each design parameter variation simulation is
from rest to 10,000 fps. This is to simulate a realistic range of narrow gauge
velocities and will be determined by a scaled thrust value to ensure that acceleration
107
is the same for each simulation. The velocity is achieved by applying a constant
acceleration of 125 Gs for 2.49 seconds. The thrust values applied to achieve
maximum velocity are shown in Table 5.10.
Table 5.10 Reduced Complexity Model Thrust Values
Case Sled Weight Value
(lbf) Thrust Values (lbf)
High 1,301 162,625.00
Typical 842 105,275.38
Low 383 47,921.63
5.3 η Force Formation
The output values of the DADS reduced complexity model simulations
consisted of forces at the four slipper beam interface locations over the simulation
time. The formation of η force comes directly from solving the translational dynamic
equilibrium equations, Equations 5.1 and 5.2, as derived from Figure 5.3 in both the
vertical and lateral directions. Two forces at each location were output from DADS
for processing in MATLAB where the data were filtered using a 10 Hz high pass
filter and then combined into a resolved force at the sled CG. The data were
windowed producing standard deviation and peak values for velocity window values
every 250 fps to ensure that locally stationary data was used in statistical calculations.
The data were compared as accomplished previously by Mixon (1971), Hooser
(1989), and Nedimovic (2004) to determine if the maximum force value over each
108
alarflfr4
1 verticaly FFFFF +++=Σ===Σ
= yiivertical FWF ηη (5.1)
left is lright isr aft is afront is f
where
alarflfr4
1lateral x FFFFF +++=Σ===Σ
= xiilateral FWF ηη (5.2)
left is lright isr aft is afront is f
where
velocity window was governed by the peak or three times the standard deviation. It
was determined that the data peaks were the governing data set as was reported by
Mixon (1971) and were subsequently used to construct the envelope function to
bound the data. The envelope function was computed using a simple algorithm that
began at a zero force value at zero velocity and searched the data set for a relative
high value that returned a greater than or equal to zero value of slope computed from
the current relative high value. Once another relative high value was found, it was
recorded and set as the current relative high value. This process was repeated until
the maximum velocity was reached. At the end of the process if no other relative
high value had been found the current relative high value was projected to the
maximum velocity point to ensure that the entire data set had been bounded. The
number of relative high values varies with each simulation result set. To make
109
Figure 5.3 Force Depiction of η Forces from Reduced Complexity Model
comparisons more direct, the envelope curve was written over the entire windowed
velocity set to make the envelope a piecewise linear function of windowed velocity as
shown for the vertical, all-typical design parameter set shown in Figure 5.4. The
MATLAB commands associated with processing the parameter variation solution
data appear in Appendix D.
A variability study was conducted to determine the variation of η force with
Vertical Force
Lateral Force
Vertical Force
Lateral Force
110
0
5000
10000
15000
20000
25000
30000
35000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Envelope CurvePeak Data
Figure 5.4 Envelope Function Depiction Vertical All-Typical Data Set
regard to random sets of equivalent rail roughness. Six solutions were accomplished
where the vertical rail roughness profiles on B- and C-Rails were shifted forward and
backward in TS respectively at 6 inch increments from typical. The first solution of
this variability study featured all of the typical values of the design parameters. The
next five solutions featured rail roughness shifting. The η force associated with these
studies was computed and the vertical and lateral results from the six solutions were
compared. The vertical force variation was bounded by a band of ±1.7 multiplied by
the standard deviation about the mean. The lateral force variation was bounded by a
band of ±1.9 multiplied by the standard deviation about the mean.
The application of η forces at the sled CG was done to ensure a conservative
approach to load distribution, not from a standpoint of a locally high value for a
111
subsequent stress analysis but from a standpoint of moment distribution. At the
HHSTT, a significant departure from previous practices was documented by Mixon
and Hooser (2002) stating that from its inception until the 1980’s, λ loading had
always been applied as a point load at the sled CG. With the advent and widespread
use of FEM software, λ loading has been applied as a distributed load across all
masses as they have been discretized based upon the model’s node distribution. The
load application difference is subtle but the resulting shear and moment distributions
across the sled structure are not. To illustrate this phenomenon consider the bounding
load cases of a point load, F, applied at the center of a beam of length L pinned at
each end and a distributed load of the same total magnitude applied to the same beam
with the same boundary conditions. The shear and moment diagrams for each case
appear in Figures 5.5 and 5.6 respectively. The distributed load case was chosen as it
represents a theoretical limit to a discretized FEM representation of a simple beam.
The maximum moment for both cases occurs at beam center and is FL*0.25 for the
point load case and FL*0.125 for the distributed load case. The point load case
produces a maximum moment 100% greater than the distributed load case. Thus η
and λ should be applied at the sled CG as to ensure proper moment calculation.
5.4 Design Parameter Variation
The reduced model was cycled through the design parameters listed in Table
Figure 3.6 where the solution of all typical parameter values was completed, followed
by solutions of different combinations of the high and low parameter values in three
112
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Length Fraction of L
Load
Fra
ctio
n of
F
Distributed Load ShearPoint Load Shear
Figure 5.5 Comparison of Point Load and Distributed Load Shear Diagrams
0
0.1
0.2
0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Length Fraction of L
Mom
ent F
ract
ion
of F
L
Distributed Load MomentPoint Load Moment
Figure 5.6 Comparison of Point Load and Distributed Load Moment Diagrams
113
steps of increasing detail. The first step consisted of individually varying the 1st
seven design parameters over their high and low values while holding all other design
parameters (except velocity) at their respective typical values resulting in 14 total
simulations. This initial step produced a comparison against typical values giving
insight into the effects of individual parameter variation. It also established an initial
parameter variation to identify any noncontributing parameters so that they could be
excluded from any subsequent parameter variations reducing the number of possible
simulations. The threshold criteria are defined as the percentage change, 10%, over
the entire velocity profile in resolved force at the CG in the vertical or lateral
direction for one parameter variation of its extremum values while all others are held
at their respective typical value. Any percentage change above the threshold denotes
a contributing parameter. Any change less than the threshold criteria denotes a non
contributing parameter. The intermediate parameter variation consisted of varying all
parameters through all possible combinations of their high and low values resulting in
128 total combinations describing the extremum space of parameter combinations.
The final parameter variation consisted of varying the remaining combinations of
remaining design parameters after applying the threshold criteria or logical simulation
number reduction from parameter dependence. The design parameter variation
served two purposes: first to identify contributing parameters, secondly to generate
data from which the dynamic load prediction tool could be constructed. A naming
convention was developed to easily detect the design parameter variation being
employed. The low value of a particular design parameter is represented by the
114
number 1, the typical value by the number 2, and the high value by the number 3.
There are eight design parameters total for each simulation. Velocity, the eighth
design parameter, is used as the independent variable in analyzing the resulting forces
so consideration of the seven other parameters is needed in the code formation. The
convention is written as a seven digit number where the left-most digit corresponds to
the seventh design parameter and the right-most digit to the first design parameter as
referenced by Table 5.7. Each digit can be a 1, 2, or 3 and represents the low, typical,
or high designation or the parameter value. Table 5.11 gives examples of convention
usage for arbitrary design parameter combinations. From this point on, design
parameter combinations will be referenced using this nomenclature.
Table 5.11 Design Parameter Combination Convention example
Design Parameter Value
7 6 5 4 3 2 1 ω Sled
Tor (Hz)
ω Sled L (Hz)
ω Sled V (Hz)
STD RR L (in)
STD RR V (in)
Slipper Gap (in)
Sled Weight
(lbf)
Convention Value
85.9 Typical
84.5 Typical
79.9 Typical
0.0147 Typical
0.0119 Typical
0.125 Typical
842.0 Typical 2222222
85.9 Typical
84.5 Typical
79.9 Typical
0.0147 Typical
0.0155 High
0.125 Typical
842.0 Typical 2222322
45.0 Low
84.5 Typical
79.9 Typical
0.0191 High
0.0083 Low
0.125 Typical
842.0 Typical 1223122
* V - Vertical; L – Lateral; Tor – Torsional; RR – Rail Roughness
115
5.4.1 Initial Design Parameter Variation
The initial design parameter variation consisted of individually varying the
seven design parameters through their high and low values, meaning that one
parameter was varied by a 1 or 3 while the other six were held at the typical value of
2. The initial parameter variation results indicated that all design parameters were
significant in either vertical or lateral force. There were instances where one force in
one direction was insignificant while the other was significant so that the parameter
variation in question could not be discarded. This result is significant in that it
validates the initial selection of design parameters and range of corresponding
variables. The all-typical design parameter case was plotted against the design
parameter variations for this initial set and includes both vertical and lateral forces on
the same plot. The resulting plots are shown in Figures 5.7-5.20.
Common to all results was the presence of a force transition point which is
defined as the departure of the η forces from λ forces, this force transition is either
higher or lower than the λ force, with the majority of the departures being lower.
Typically the transition occurred between 800 and 2000 fps. Also shown was the
variation of the ratio of lateral to vertical dynamic forces which has been traditionally
held to 0.6. These phenomena are documented by Mixon and Hooser (2002, Figure
16) and Hooser (1989, Figure 4.5) for a wide array of sled configurations representing
a correspondingly wide array of sled design parameters. Error tracking did not
indicate that any of the simulations were out of tolerance.
116
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.7 Initial Parameter Variation Vertical Force Results: Mass
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.8 Initial Parameter Variation Lateral Force Results: Mass
117
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.9 Initial Parameter Variation Vertical Force Results: Slipper Gap
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.10 Initial Parameter Variation Lateral Force Results: Slipper Gap
118
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.11 Initial Parameter Variation Vertical Force Results: Vertical Rail
Roughness
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.12 Initial Parameter Variation Lateral Force Results: Vertical Rail
Roughness
119
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.13 Initial Parameter Variation Vertical Force Results: Lateral Rail
Roughness
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.14 Initial Parameter Variation Lateral Force Results: Lateral Rail Roughness
120
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.15 Initial Parameter Variation Vertical Force Results: Vertical Natural
Frequency
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.16 Initial Parameter Variation Lateral Force Results: Vertical Natural
Frequency
121
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.17 Initial Parameter Variation Vertical Force Results: Lateral Natural
Frequency
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.18 Initial Parameter Variation Lateral Force Results: Lateral Natural
Frequency
122
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)Typical Vertical
Low Vertical
High Vertical
Lambda Vertical
Figure 5.19 Initial Parameter Variation Vertical Force Results: Torsional Natural
Frequency
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (fps)
Forc
e (lb
f)
Typical Lateral
Low Lateral
High Lateral
Lambda Lateral
Figure 5.20 Initial Parameter Variation Lateral Force Results: Torsional Natural
Frequency
123
5.4.2 Intermediate Design Parameter Variation
The intermediate design parameter variation was accomplished to define the
extremum space of the design parameters and attempt to encompass the anticipated
peak values of the vertical and lateral η forces. The subsequent formation of a design
tool would, at a minimum, have to include the all-typical, initial, and intermediate
solutions to be able to interpolate a desired point from within the design parameter
space. As a result, the intermediate design parameter variation consisted of varying
all combinations of the high and low values for all seven design parameters resulting
in 128 total solutions as shown in Table 5.12. The simulation index consisted of all
binary combinations of a seven digit number using 1s and 3s. Analysis of these
results consisted of a cursory visual comparison to the typical values and was deemed
acceptable based on the performance of the initial parameter variation results and that
error tracking did not indicate that any of the simulations were out of tolerance.
Table 5.12 Intermediate Design Parameter Variation Values Design
Parameter 7 6 5 4 3 2 1
1 1 1 1 1 1 1 Value
3 3 3 3 3 3 3
5.4.3 Final Design Parameter Variation
The final design parameter variation study was accomplished to give greater
detail to areas of interest bounded by the intermediate design parameter and initial
124
design parameter studies. The total number of possible simulations was 2187 or 37.
Instead of running all possible combinations it was decided that a reduced set of 192
total simulations would be used in this final stage. This reduced set was defined by
analyzing the initial design parameter variations results and then applying a bounding
influence technique. This technique was applied by visually analyzing Figures 5.7-
5.13 individually and then determining which values of the design parameters were
influential in either the maximum or minimum values of the corresponding η force.
It was considered absolutely necessary to consider the vertical and lateral forces
simultaneously as the bounding design parameters for each force were not always the
same. An example of this phenomenon is best illustrated in Figure 5.10 where the
variation of rail roughness is shown. Above 6,000 fps there is a significant increase
in the η force in the vertical direction for a low lateral rail roughness value and above
8,000 fps there is significant in crease in the η force in the lateral direction for a high
lateral rail roughness value. The lower parameter value bounds the maximum vertical
values while the higher parameter bounds the maximum value for the lateral force
direction. After a careful study of the figures, the bounding design parameter values
were determined and are shown in Table 5.13. Τhe inclusion of all three parameter
values for design parameter 7 indicates that the complete influence of the design
parameter values and that the η curves are intertwined as shown in Figure 5.13. From
the set of 192 total simulations it was found 32 had been previously solved so 160
unique solutions were required to complete this task. Error tracking did not indicate
that any of the results were out of tolerance.
125
Table 5.13 Final Design Parameter Variation Values Design
Parameter 7 6 5 4 3 2 1
1 1 1 1 1 1 1
2 2 2 3 3 3 3 Value
3 N/A N/A N/A N/A N/A N/A
126
6. DEVELOPMENT OF CORRELATION OF PREDICTED LOAD AND DESIGN PARAMETERS
The ultimate goal of this study is the production of a design tool which can
effectively correlate the input variables to the output forces so that the designer may
effectively apply the design tool during the sled design process. The functional
relationship sought is represented for a given η force in Equation 6.1. This was
accomplished by correlating the design parameters to the η forces produced by the
efforts documented in Chapter 5. Several correlation methods were studied including
linear multivariate least squares regression, nonlinear multivariate least squares
regression, nondimensional analysis, and multivariate interpolation. The motivation
behind the studying of regression methods was the possibility of arriving at a closed
form solution which may be applied either by hand or within a simulation
environment in a variety of applications whereas an interpolation method is
somewhat fixed as the algorithm must be programmed. Since the design tool will
ultimately be employed in an environment where its function must be expedient, any
implementation must be acutely representative of the data produced in Chapter 5 and
easily employed so that it will not suffer the same fate as that of SIMP and
SLEDYNE.
),,,,,,,( 12345678 DPDPDPDPDPDPDPDPfF =η (6.1)
127
6.1 Linear Least Squares Regression
To begin the correlation effort a basic multivariate least squares regression
was accomplished on the η force data set. The assumed solution took the form shown
in Equation 6.2. The regression coefficients were solved using Equation 6.3 by
inverting the summation matrix and multiplying it by the right hand side summation
vector (Volk 1958). The regression coefficients are shown in Table 6.1. The
correlation coefficient, r2, was calculated using Equation 6.4 and yielded the values of
0.82 and 0.76 for the vertical and lateral η forces respectively. These values, both
less than 0.99, indicate an inadequate regression which was due in part to the
piecewise linear nature of the η force over velocity and nonlinear variation of other
design parameters. The motivation behind this effort was to establish a basis from
which to launch a nonlinear regression study to investigate a closed form data
correlation. The individual correlation coefficients, r, were calculated using Equation
6.5 and are shown in Table 6.2 for each of the design parameters. From the results in
Table 6.2 it is obvious that the linear correlation is poor for design parameters mass,
slipper gap, rail roughness lateral (vertical only), vertical, lateral and torsional sled
natural frequency and better for design parameters rail roughness vertical, rail
roughness lateral (lateral only), and velocity. This information is useful in
determining in how to vary the assumed curve fit for a nonlinear regression analysis
since higher correlation coefficients imply that the respective variable is better
represented by the method used.
128
... 88332211 xbxbxbxbay +++++= (6.2)
1,...,8index theis itscoefficien regression are ba,
valuesparametersdesign theare force variable,dependent theisy
where
i
ixη
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∑
∑∑∑
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
∑∑∑∑∑
∑∑∑∑∑∑∑∑∑∑∑∑∑∑
yx
yxyx
y
b
bba
xxxxxxxx
xxxxxxxxxxxxxxxx
xxxxN
8
2
1
8
2
1
283828188
823222212
813121211
8321
......*
.....................
...
...
...
(6.3)
points data ofnumber total theis Nwhere
''
2
22
∑∑=
yc
r (6.4)
forceoutput isy valuesparametersdesign theare
y ofmean theis y
)y-(yy'
))((`
yx'...yx'yx'c'where
ii
22
8822112
i
iii
x
yyxxyx
bbb
∑∑∑ ∑
∑∑∑∑
=
−−=
+++=
)''(
'5.022∑ ∑
∑=yx
yxr
i
iyxi
(6.5)
∑∑∑∑
=
=2
ii2
i
iii
)x-(xx'
)y-(y)x-(xyx'
where
129
1,...,8index theis iforceoutput isy
valuesparametersdesign theare xofmean theis x ii
ix
Table 6.1 Linear Multivariate Least Squares Regression Coefficients
Force Direction Regression Coefficients
Design Parameter
Vertical Lateral
a -16049.8 -9628.8
b1 Sled Mass 2.26 1.98
b2 Slipper Gap -75754.7 -40411.0
b3 Vertical Rail Roughness 1997732.8 574909.9
b4 Lateral Rail Roughness 48638.5 471002.2
b5 ωVertical -1.28 0.29
b6 ωLateral 1.40 9.59
b7 ωTorsional 12.65 3.85
b8 Velocity 4.06 2.21
130
Table 6.2 Linear Multivariate Least Squares Correlation Coefficients
Individual Correlation Coefficient Values, rRegression Coefficients
Design Parameter
Vertical Lateral
b1 Sled Mass 0.07 0.01
b2 Slipper Gap -0.07 -0.07
b3 Vertical Rail Roughness 0.45 0.24
b4 Lateral Rail Roughness 0.01 0.24
b5 ωVertical -0.00 0.01
b6 ωLateral 0.01 0.06
b7 ωTorsional 0.04 0.03
b8 Velocity 0.77 0.79
6.2 Nonlinear Least Squares Regression
The nonlinear least squares regression was studied to better characterize the η
force curves and to investigate a closed form relationship between the design
parameters and the output η forces. A nonlinear regression analysis gives rise to a
vast number of possible relationships of design parameters and assumed curve fits.
For the current study the nonlinear forms were limited to polynomial or power forms.
131
6.2.1 Polynomial Form
The polynomial form was implemented to explore the correlation of different
exponent values in a linear combination of design parameters or by using exponent
values and nonlinear combinations of design parameters. The first part of this study
focused on combinations of different exponent values for the polynomial while the
second part focused on different nonlinear combinations of variables.
In the first part the exponent values were cycled through the range of 1, 2, and
3 for all possible values (high, typical, and low) for all eight design parameters in the
form of Equation 6.6. The individual correlation coefficients for each prediction were
tracked to determine if a particular fit was acceptable. After cycling the exponents
through the prescribed range it was found that none of the individual correlation
coefficients reached an acceptable value as indicated by their ranges shown in Table
6.3.
... 832188332211nnnn xbxbxbxbay +++++= (6.6)
1,...,8index theis iexponents theare n
tscoefficien regression are ba, valuesparametersdesign theare
force variable,dependent theisy where
i
i
ixη
The second part of the nonlinear regression effort utilized single variable
combinations of all possible combinations of the design parameters of the form given
in Equation 6.7. Following the least squares methodology depicted in Equation 6.3
132
Table 6.3 Polynomial Individual Correlation Coefficient Ranges Design Parameter DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8
High Value 0.11 0.08 0.45 0.25 0.02 0.06 0.04 0.79
Low Value 0.07 0.07 0.24 0.00 0.00 0.00 0.03 0.67
the initial formation shown in Equation 6.7 yielded correlation coefficients, r2, of 0.71
vertical and 0.71 lateral, showing that this correlation was better but still not within an
acceptable range. Equation 6.7 was modified slightly by changing the beginning
index of j from i to 1 which resulted in Equation 6.8. The slight change in indices
adds a factor of 2 to the xixj terms where i≠j. This modification significantly
improved the correlation coefficients to 0.90 vertical and 0.82 lateral. This change
was the first regression based effort to yield acceptable r2 values that included
contributions from all eight design parameters. Studying the individual correlation
coefficients, it was seen for the vertical direction that variables containing the design
parameters rail roughness vertical and velocity were influential in this form. For the
lateral direction variables containing design parameters mass, rail roughness vertical,
rail roughness lateral, and velocity were influential in this form. The r2 values near
0.9 indicate a positive representation of the majority of η data but fail to represent all
individual cases. In some cases, η force is under predicted by as much as 50% thus
disqualifying this regression method.
8877665544332211
8
1
8
xbxbxbxbxbxbxbxbxxayi ij
ji +++++++++= ∑∑= =
(6.7)
133
8877665544332211
8
1
8
1
xbxbxbxbxbxbxbxbxxayi j
ji +++++++++= ∑∑= =
(6.8)
6.2.2 Power Form
The power form was implemented as shown in Equation 6.9. The exponent
values were solved by taking the log of both sides, implementing a least squares
regression of the logarithmic values, and then solving for the exponent values. The
least squares method was performed on Equation 6.10 and yielded the exponent
values shown in Table 6.4. R2 values for this form were 0.99 vertical and 0.98 lateral.
The resulting correlation coefficient values associated with each design parameter are
shown in Table 6.5 and prove to be excellent for velocity, but unacceptable for other
parameters. The high correlation values for velocity are partially due to the fact that
the method automatically forces the prediction of 0 force at 0 velocity. There is some
significant deviation of the prediction from the η extremum values for some cases
(see Figures 6.1 and 6.2). Thus this method is unacceptable for use as a correlation
tool in the current study although it was the most accurate direct linear least squares
regression method.
876543218765432187654321bbbbbbbbabb
TORbL
bV
bL
bV
bba xxxxxxxxeVRRRRSGWeF == ωωωη (6.9)
)log()log()log()log()log()log()log()log()log(
88776655
44332211
xbxbxbxbxbxbxbxbaF
+++
+++++=η (6.10)
134
Table 6.4 Power Form Exponent Values
Force Direction Regression Exponent Values
Design Parameter
Vertical Lateral
a 36.91 16.07754
b1 Sled Mass 0.1356 0.18275
b2 Slipper Gap -0.255 -0.3363
b3 Vertical
Rail Roughness
0.9885 0.66742
b4 Lateral Rail Roughness 0.0438 0.42874
b5 ωVertical -0.01009 0.00366
b6 ωLateral 0.00588 0.0616
b7 ωTorsional 0.0625 0.05653
b8 Velocity 1.083 1.0708
6.3 Nondimensional Analysis
Nondimensional analysis is a method which employs groupings of dependent
and independent variables to examine meaningful relationships among all of the
variables in a nondimensional form. This method is employed extensively in the
thermal fluids area of study where well known dimensionless parameters such as
Reynolds number and Nusselt number have been developed using nondimensional
135
Table 6.5 Power Form Individual Coefficient Values
Individual Correlation Coefficient Values, r Regression Coefficients
Design Parameter
Vertical Lateral
b1 Sled Mass 0.03 0.04
b2 Slipper Gap -0.01 -0.01
b3 Vertical
Rail Roughness
0.11 0.07
b4 Lateral Rail Roughness 0.02 0.05
b5 ωVertical -0.00 0.00
b6 ωLateral 0.00 0.12
b7 ωTorsional 0.01 0.01
b8 Velocity 0.99 0.98
techniques specifically the Buckingham Pi (Π) Theorem. The Π Theorem outlined
by White (1994) will be used in this portion of the study. The application of the Π
theorem to solid mechanics is not widespread but was employed recently in
Smzerekovsky’s (2004) study of hypervelocity impact. The Π theorem in general
consists of evaluating the function and number of dependent and independent
variables and then listing the dimensions of each variable. Applying this to the
function in question, Equation 6.1, the variables and associated dimensions are shown
136
0
10000
20000
30000
40000
50000
60000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)Eta Force VerticalLogarithmic Prediction
Figure 6.1 2212222 Vertical Best Power Form Prediction vs η Force
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)
Eta Force VerticalLogarithmic Prediction
Figure 6.2 1111331 Vertical Worst Power Form Prediction vs η Force
137
in Table 6.6. There are 9 variables and choosing a dimensional system of mass, M,
length, L, and time, T, there are 3 dimensions. Following White’s (1994) treatment
there should be 6 distinct Π parameters. Using the repeatable groupings of Velocity,
Weight, and Slipper Gap, the 6 Πs are written in Equation 6.11 and represent
essentially nondimensional values of design parameters and output force. It is
interesting to note that from the initial collection of 1 dependent variable and 8
independent variables that there are now 6 Π values that represent 1 dependent
variable and 5 independent variables which is a reduction in the amount of original
variables. The exponential values for each Π were next solved to satisfy the
nondimensional requirement, the results of which are presented in Table 6.7 and
applied in Equation 6.12. At this step in the process, a relationship is sought between
the dependent variable, Π1, and the independent variables, Π 2-6, shown in Equation
6.13 and is analogous to Equation 6.1. Beginning with the initial relative success of
the power form the relationship in Equation 6.14 was pursued. The particular
formation was developed to satisfy several known aspects of the data; velocity and
vertical rail roughness are proportional to the vertical output force while slipper gap is
inversely proportional to vertical output force. The resulting relationship returned a
spread of data that proved to be too wide even after graphical representation with
exponential scaling. The best result in each respective force application direction is
presented in Figures 6.3 and 6.4. The resulting data spread represented a situation
where a closed form mathematical relationship was not possible. Other power
combinations were pursued that featured cubic powers of velocity and quintic powers
138
of slipper gap whose results met with the same unacceptable spread of data. The next
relational attempt was to run the nondimensional parameters through the same gamut
of least squares regression methods used in the preceding sections. The best attempt
was the logarithmic regression which produced r2 values of 0.54 vertical and 0.29
lateral, thus representing an inadequate regression of the non dimensional values.
Table 6.6 Design Parameter Dimensional Analysis Design Parameter Output DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8
Variable Fη W SG RRV RRL ωV ωL ωTOR V
Dimension MLT-2 MLT-2 L L L T-1 T-1 T-1 LT-1
000126
000125
000124
00023
00022
000221
TLMTLTLMTLSGWV
TLMTLTLMTLSGWV
TLMTLTLMTLSGWV
TLMLLTLMTLRRSGWV
TLMLLTLMTLRRSGWVTLMMLTLTLMTLFSGWV
rqqqppTOR
rqp
onnnmmL
onm
lkkkjjV
lkj
ikhhggL
ihg
feeeddV
fed
cbbbaacba
===Π
===Π
===Π
===Π
===Π
===Π
−−−
−−−
−−−
−−
−−
−−−
ω
ω
ω (6.11)
Table 6.7 Π Exponent Values
Π1 Π2 Π3 Π4 Π5 Π6
0=a 0=d 0=g 1−=j 1−=m 1−=p
1−=b 0=e 0=h 0=k 0=n 0=q Exponent Values
0=c 1−=f 1−=i 1=l 1=o 1=r
139
VSG
VSG
VSGSGRRSGRRWF
TOR
L
V
L
V
ω
ω
ω
=Π
=Π
=Π
=Π
=Π
=Π
6
5
4
3
2
1
(6.12)
),,,,( 654321 ΠΠΠΠΠ=Π f (6.13)
SGRRVRR
TORLL
VV
ωωω
=ΠΠΠ
ΠΠ=Π
653
421 (6.14)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 50 100 150 200 250 300 350 400 450 500
(π2π4/π3/π5/π6)^(1/19)
π 1^(
1/10
))
Figure 6.3 Vertical Π Theorem Analysis
140
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 50 100 150 200 250 300 350 400 450 500
(π2π4/π3/π5/π6)^(1/19)
π 1^(
1/10
))
Figure 6.4 Lateral Π Theorem Analysis
6.4 Multivariate Interpolation
Multivariate interpolation is a method of computing a correlated value from a
multidimensional data table of input design parameters and associated η force output
values. The point of interest is bounded by input variables higher and lower in value.
The output values associated with each set of input variables is then used to linearly
interpolate the value of the point of interest to produce the desired output value.
Multivariate interpolation is basically a local application of linear regression as the
point of interest is a weighted average of the adjacent bounding points. On the other
hand, a regression method is influenced by values of all points in the data. A
multivariate interpolator meticulously considers weighted averages between distinct
points in a data table where a regression equation would be applicable to all points
141
throughout the table. Multivariate interpolation, if properly applied, is accurate for
almost all data types since it is bounded while regression techniques seek to satisfy an
assumed mathematical relationship for the entire data set. In the current study, the
nonlinear nature of the η force data makes multivariate regression the correct choice
for the η force prediction tool.
For clarity, two examples are presented. The first consists of a 2-dimensional
example that requires four sets of data points to bound the desired point. The high
values will be designated by a 1 and the low values by a 0 and the interpolated values
by 0.5. There are output values associated with each data point set. The desired point
is designated by a value of (0.5,0.5) with the 2-dimensional example displayed in
Figure 6.5. Pairs of data point sets are interpolated as shown by the interpolation
points which in turn are interpolated to return the desired point designated in the
figure by a square. Equation 6.15 was applied to the pairs of data points to perform
the interpolation. As applied and conducive to the piecewise linear nature of the data,
the interpolation method was implemented as a two dimensional recursive
interpolation.
A second example, 3-dimensional, appears in Figure 6.6 and demonstrates the
extension of the 2-dimensional example with the addition of another dimension and
by the transformation of the interpolation space from a square to a cube. The
interpolation is accomplished using Equation 6.15 and is easily adapted for use by
considering pairs of adjacent points and stepping towards the desired location. Again
the high values are designated by a 1 and the low values by a 0. The desired point is
142
-0.5
0
0.5
1
1.5
-0.5 0 0.5 1 1.5
Dimension 1
Dim
ensi
on 2
Data SetsDesired PointInterpolation Points
(0,1,1.5)
(0.5,0.5,1.56)
(1,0.5,1.875)(0,0.5,1.25)
(0,0,1.0) (1,0,2.0)
(1,1,1.75)
Figure 6.5 2-Dimensional Interpolation Example Depiction
hhdesiredlh
lhdesired yxx
xxyy
y +−⎟⎟⎠
⎞⎜⎜⎝
⎛−−
= )( (6.15)
(0.5, 0.5, 0.5) and is ultimately found to be 1.88. The interpolation process may be
performed between any eight points; in this study, it was more efficient to perform
the interpolation between adjacent points which are for this example the circles and
then interpolated to the points designated by the Xs. These four points were then
143
interpolated to the two hexagons and finally to the square; this process is detailed in
Table 6.8. The total number of bounding points for the three dimensional case is 8
points indicating a trend of 2n where n is the number of dimensions or parameters
needed to bound the desired point.
Figure 6.6 3-Dimensional Interpolation Example Depiction
(1,1,1,2.5)
(0,1,1,2.0)(0,0,1,2.0)
(0,1,0,1.25)
(1,1,0,1.75)(1,0,0,1.5)
(0,0,0,1.5)
(1,0,1,2.5)
(0.5,0.5,0.5,1.88)
Dimension 1
Dimension 2
Dimension 3
144
Table 6.8 Three Dimensional Interpolation Detail
Dimensions
(X1,X2,X3,Y) (X1,X2,X3,Y) (X1,X2,X3,Y) (X1,X2,X3,Y)
(0,0,0,1.5)
(0,0,1,2.0) (0,0,0.5,1.75)
(1,0,0,1.5)
(1,0,1,2.5) (1,0,0.5,2.00)
(0.5,0,0.5,1.875)
(0,1,0,1.25)
(0,1,1,2.0) (0,1,0.5,1.625)
(1,1,0,1.75)
(1,1,1,2.5) (1,1,0.5,2.125)
(0.5,1,0.5,1.875)
(0.5,0.5,0.5,1.875)
Circle X Hexagon Square
Symbol
Extending the interpolation to higher dimensions, ultimately 8, a systematic
approach is warranted due to the many calculations needed to perform data sorting
and adjacent point interpolation. The total number of data points needed to bound the
desired point is 28 or 256. The interpolation algorithm was implemented on the data
set produced in Chapter 5 in a Microsoft© Excel spreadsheet and was found to be
145
accurate and expedient. The shortest runtimes for the spreadsheet were 20 seconds to
return dynamic loads for a full 8 dimensional design parameter set. The spreadsheet
functioned such that if a design parameter set coincided with a η force database point
then the vertical and lateral η force database values were returned. Further testing
showed that the spreadsheet properly interpolated η force values for single parameter
variations.
A concern when using an interpolation method is that representing a design
parameter set outside the range of the original data set requires extrapolation. To
alleviate this concern, the maximum velocity range was linearly extrapolated to
10,250 fps from the original value of 10,000 fps. Slipper gap values should not
exceed the range specified by regulation (846th Test Squadron 2006) and rail
roughness is always be adequately bounded by current design parameter range based
on actual measurements. A remaining concern is the inclusion of natural frequency
of the sled since it is difficult to predict the values produced during a sled
development cycle. The typical range expected from the reduced complexity model
was taken directly from the LSR sled with no payload since this design is the most
likely candidate to experience high velocity ranges for the foreseeable future.
However the range of natural frequencies studied herein is not all encompassing.
Thus, the question remains: how are natural frequency values outside of this range to
be accounted for? In the event that the desired design parameter values were in the
unbounded space, the highest adjacent value was returned in place of the interpolated
value and the user was notified of such. For situations where η limits are greatly
146
exceeded, or completely unbounded the most logical and conservative approach was
to prescribe the maximum η force value and notify the user. It is seen that for almost
all-typical design parameter values that η forces were less than their λ or SIMP
counterparts. When η force design parameter ranges were exceeded and extreme
conservatism was desired then the next most accurate force prediction tool was λ or
SIMP. However, given the inaccurate nature of these two methods it would be
advisable to use only λ with the condition that the answer would be highly
conservative for values of typical rail roughness only.
147
7. COMPARISON OF PREDICTED LOAD TO PREVIOUS LOAD PREDICTION METHODS
The η force generated in the course of this study was compared to methods
commonly used to predict design loads including λ, SIMP, and SLEDYNE. The
comparison was conducted in two parts, a general comparison of force magnitude
predicted with the various techniques and then a comparison of sled designs produced
using the different methods to evaluate the effectiveness of the new prediction
method. SLEDYNE was successfully recompiled in a Microsoft® Windows
environment; however, the required COSMIC NASTRAN® input files could not be
generated for the comparison study, and no previously designed narrow gauge sleds
using SLEDYNE were located. These complications made SLEDYNE unusable for a
comparison tool but since it is a detailed FEM implementation of the same method
used to develop SIMP, the comparison is made for both within the context of SIMP.
7.1 Force Magnitude Comparison
The comparison of the magnitudes of design forces predicted by η, λ, and
SIMP was conducted to determine the accuracy of each method and to develop
familiarity comparing η forces to those from more well known methods. SIMP and λ
force values were calculated alongside η force values during parameter variation data
processing to give an accurate comparison. In the majority of the cases studied, for
typical values of rail roughness, SIMP and λ forces were larger in magnitude than η
forces after the force transition point is reached (see Section 5.4.1). There are a few
148
cases where η forces exceed λ and SIMP forces, these typical rail roughness cases
and associated values shown in Table 7.1. It is seen in Table 7.1 that in the situation
when one force direction of η exceeds λ or SIMP, the other direction tends to be
lower leading to sled designs which may ultimately contain lower mass. The ratio of
lateral to vertical force as dictated in the formation of λ, typically 0.6, is not observed
in the case of η where the ratio varies from 0.07 to 1.8 and is dictated by velocity, rail
roughness and slipper gap. The lateral to vertical force ratio for η forces is directly
related to the mechanisms that cause η lateral and η vertical to increase or decrease.
The same study of SIMP derived forces yields a range of 0.22 to 0.28; however, the
lateral values of SIMP generated forces did not correlate well to test data and
subsequently tend to under predict lateral forces (Mixon 1971). In studies of LSR
and CO test data (Turnbull and Minto 2003, Leadingham and Schauer 2001) the ratio
of lateral to vertical force varies and ranges from 0.07 to 1.4.
It was found during the design parameter variation study of η that
combinations of certain design parameters yielded higher η forces. These sets were
not the same for the vertical and lateral force directions. The maximum η force
values are shown in Table 7.2 and occurred at design parameter values of 3123313
vertical and 3313313 lateral. It was documented previously and verified currently
that rail roughness and slipper gap are highly influential in the magnitude of dynamic
loading. As can be seen in Table 7.2, the high values of rail roughness, and the lower
value of slipper gap return the maximum η forces at maximum velocity. Maximum
force occurring at maximum velocity had been noted from the earliest studies of
149
dynamic loading (ISTRACON Handbook 1961, Mixon 1971).
Table 7.1 Selected η Comparisons to λ and SIMP at 10,000 fps
η Peak
λ Peak
SIMP Peak
Case Vertical
(lbf) Lateral
(lbf) Vertical
(lbf) Lateral
(lbf) Vertical
(lbf) Lateral
(lbf)
2222221 28,158 26,348 30,670 18,402 48,548 11,329
2222223 36,740 17,689 104,080 62,448 47,923 10,414
1222222 48,516 19,379 67,376 40,426 48,357 10,771
Table 7.2 Maximum η Force Values at 10,000 fps
Case η Vertical
(lbf) η Lateral
(lbf)
3123313 89,094 22,379
3313313 59,669 53,414
7.1.1 Vertical Force Comparison
At higher sled velocities, the comparison of the vertical η forces to those
predicted by SIMP and λ follow the general trend that higher values of vertical rail
roughness standard deviation yielded high η forces than the corresponding values of
SIMP and λ. This is due to the choice of the high value of vertical rail roughness,
30% greater than surveyed, and to the fact that SIMP and λ are derived from typical
150
rail roughness conditions and do not account for changes in rail roughness deviation.
The best example is shown in Figure 7.1 where η vertical 1111331 and 1111131 are
plotted against corresponding λ and SIMP values. Intuitively smaller slipper gap,
high vertical sled natural frequency, and smaller mass amplify the effect of high
vertical rail roughness. This is a natural conclusion since λ and SIMP are derived for
fixed slipper gaps and depend directly on mass to calculate loading. The mass
variation effect is less in SIMP than in λ since with λ it is a linear relationship and
with SIMP it is a Mass0.25 term. The vertical sled natural frequency effect is less in
SIMP than in λ since SIMP can account for some flexibility variation. When all
parameters are typical, 2222222, η is less than λ and SIMP as shown in Figure 7.2.
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)
1111331 Vertical Eta1111131 Vertical Eta1111331 Vertical SIMP1111131 Vertical SIMPVertical Lambda
Figure 7.1 Vertical Force Extreme Case, 1111331, Comparison
151
0
10000
20000
30000
40000
50000
60000
70000
80000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)
2222222 Vertical Eta2222222 Vertical SIMP2222222 Vertical Lambda
Figure 7.2 Vertical Force Typical Case, 2222222, Comparison
7.1.2 Lateral Force Comparison
The comparison of η forces in the lateral direction to those generated from
SIMP and λ follow the similar general trends reported in the study of the vertical
force comparison. Standard deviation of lateral rail roughness was the primary
driving factor behind the deviation of η from SIMP and λ. Increases in slipper gap
tended to lessen the effect and conversely decreases in slipper gap tended to amplify
the effect. Increases in sled velocity caused increases in η forces as was the case with
λ and SIMP. Combinations of high natural frequencies and high rail roughness
tended to increase η loading. The amplification effect was similar to that seen for
lower and higher gap. There was significant coupling of the lateral and vertical
loading in the sense that lateral η was sensitive to changes in vertical sled parameters
152
such as vertical rail roughness values and vertical sled natural frequency. The
coupling trend was more evident than was observed in the vertical direction.
In general, η was lower than λ for typical and low values of rail roughness as
shown in Figure 7.3 for the extreme case of 1111133. Since λ depends directly on
sled mass but does not consider rail roughness, λ was less than η for low mass values
as shown in Figure 7.4 for the extreme case of 2222221. In general, SIMP was
significantly less than η for all higher velocity cases studied. This substantial result is
due to SIMP not being correlated to lateral test data since it did not consider lateral
rail survey data and did not accurately characterize lateral slipper rail contact. It is
seen in hypersonic test data from the LSR test (Minto and Turnbull 2003) that lateral
and vertical dynamic effects are similar in magnitude. It should be remembered that
λ and SIMP are fixed in many of their parameters when compared to η. In the all-
typical, 2222222, case comparison, Figure 7.5, η proves to be lower at every velocity
other than 1,500 fps where η slightly exceeds λ. η is lower than SIMP for velocities
under 4000 fps.
7.2 Sled Design Comparison
To further illustrate the comparison among the force prediction methods a
mock sled design was performed to accentuate the differences between η, λ, and
SIMP. The reduced complexity model narrow gauge was chosen as the sled
configuration for consideration. The slipper beam material is normalized 4130 steel
(Department of Defense 2003) whose properties are given in Table 7.3 while the sled
153
0
10000
20000
30000
40000
50000
60000
70000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)1111133 Lateral Eta1111133 Lateral SIMP1111133 Lateral Lambda
Figure 7.3 Lateral Force Extreme Case, 1111133, Comparison
0
5000
10000
15000
20000
25000
30000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)
2222221 Lateral Eta2222221 Lateral SIMP2222221 Lateral Lambda
Figure 7.4 Lateral Force Extreme Case, 2222221, Comparison
154
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Velocity (ft/s)
Forc
e (lb
f)2222222 Lateral Eta2222222 Lateral SIMP2222222 Lateral Lambda
Figure 7.5 Lateral Force Typical Case, 2222222, Comparison
Table 7.3 Slipper Beam Material Specification Young’s Modulus
(psi) Poisson’s
Ratio Density (lbf/in3)
Yield Stress (psi)
Ultimate Stress (psi)
29E6 0.32 0.283 70E3 90E3
body design material is a composite whose properties are given in Table 7.4. The
comparison to be made here was more than a basic ratio of loading for η and SIMP
since it will take into consideration the mass distribution which is dictated by the
capacity to successfully resist the resulting load distribution within the structure. The
comparison was further defined by setting the parameters to their typical values from
the η force formation study as shown in Table 7.5.
155
Table 7.4 Sled Body Material Specification Young’s Modulus
(psi) Poisson’s
Ratio Density (lbf/in3)
Yield Stress (psi)
Ultimate Stress (psi)
11.8E6 0.28 0.067 60E3 60E3
Table 7.5 Design Parameters for Mock Sled Design Design
Parameter DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8
Variable W (lbf)
SG (in)
RRV (in)
RRL (in)
ωV (Hz)
ωL (Hz)
ωTOR (Hz)
V (fps)
Value 842.0 0.125 0.0119 0.0147 79.9 84.5 85.9 9,500
The goal of the design comparison was to conduct a concept feasibility-level
design of the sled structure as depicted in Figure 7.6 to minimize weight and optimize
vonMises stress within the sled body and slipper beams. Initial study dimensions are
listed in Table 7.6 and initial weight detailed in Table 7.7. The stress analysis was
limited to basic beam analyses of the sled body and slipper beam and did not entail a
detailed study of their connections or local three dimensional effects. The stress
values were maximum values of beam bending, axial and shear resolved into their
corresponding vonMises values. To further simplify the process, the thickness of the
sled body and slipper beams were the varied parameters. Indirectly this caused shifts
in weight and thus resulted in design iterations. Subject to the desire of the analyst,
these iterations were reasonably curtailed in the present study. A Factor of Safety
(FS) of 1.0 was applied to the yield stress and used to compute a Margin of Safety
(MS) as shown in Equation 7.1 (Krupovage, Mixon, Bush 1991).
156
Figure 7.6 Design Study Sled Structure
Table 7.6 Initial Study Dimensions
Parameter Slipper Beam
Height (in)
Slipper Beam
Width (in)
Slipper Beam
Thickness (in)
Sled Body Radius
(in)
Sled Body Thickness
(in)
Value 3.5 3.5 0.375 8.06 1.354
Table 7.7 Initial Study Weight Detail
Component Sled Body
Slipper Beam
(2 total) Slippers (4 total) Miscellaneous Total
Weight (lbf)
573.856 95.513 132.00 40.631 842.00
Sled Body
Slipper Beam
Slipper
157
FS
MS
YieldAllowable
Design
Allowable
σσ
σσ
=
−= 1 (7.1)
stress yield material theis where
Yieldσ
7.2.1 λ Design Study
The λ mock sled design considers only weight and velocity to form the λ
force associated with the sled design. The method of calculating the λ force
(Krupovage et al. 1991) is shown in Equation 7.2 for a narrow gauge sled and
produced initial force values of 63,992 lbf vertical and 38,395 lbf lateral utilizing the
parameters given in Table 7.8. During the course of design iteration, it was apparent
that adding material to the slipper beam to withstand the vonMises stress levels
caused a linear load increase while the stress levels decreased cubically. The final
study dimensions are shown in Table 7.9 and correspond to a total sled weight of
847.75 lbf which is an increase of 5.75 lbf over the initial design. The resulting shear
and moment diagrams for the slipper beam and sled body appear in Figures 7.7-7.10.
This design resulted in a MS of 0.0 at the forward slipper beam midpoint with final
design loads of 66,330 lbf vertical and 39,798 lbf lateral.
VerticalLateral
Vertical
FFWVWF
λλ
λ λ*6.0
**008.0=
== (7.2)
velocityis weightsled is
0.008 factor, gage narrow theis where
VW
λλ
158
Table 7.8 λ Initial Design Parameters λ Design Parameter
Sled Weight
(lbf) Velocity
(fps) λ factor
842.0 9,500 0.008
Table 7.9 λ Final Study Dimensions
Parameter Slipper Beam
Height (in)
Slipper Beam
Width (in)
Slipper Beam
Thickness (in)
Sled Body Radius
(in)
Sled Body Thickness
(in)
Value 3.5 3.5 0.598 8.06 1.25
-20000
-15000
-10000
-5000
0
5000
10000
15000
20000
0 0.2 0.4 0.6 0.8 1
Length fraction of L=26.04 (in)
Forc
e (lb
f)
VerticalLateral
Figure 7.7 λ Slipper Beam Shear Diagram
159
-250000
-200000
-150000
-100000
-50000
0
0 0.2 0.4 0.6 0.8 1
Length fraction of L=26.04 (in)
Mom
ent (
in lb
f)
VerticalLateral
Figure 7.8 λ Slipper Beam Moment Diagram
-40000
-30000
-20000
-10000
0
10000
20000
30000
40000
0 0.2 0.4 0.6 0.8 1
Length fraction of L=120.67 (in)
Forc
e (lb
f)
VerticalLateral
Figure 7.9 λ Sled Body Shear Diagram
160
-4500000
-4000000
-3500000
-3000000
-2500000
-2000000
-1500000
-1000000
-500000
0
0 0.2 0.4 0.6 0.8 1
Length fraction of L=120.67 (in)
Mom
ent (
in lb
f)
VerticalLateral
Figure 7.10 λ Sled Body Moment Diagram
7.2.2 SIMP Design Study
The SIMP force calculation was completed by applying the SIMP process to
the design under consideration. The initial SIMP parameters appear in Table 7.10
and served as the starting point for this design study. After iteration, the final SIMP
design parameters, Table 7.11, were produced and applied as an illustration of the
load determination accomplished in 7 steps (Krupovage et al. 1991) for the current
design. The first step of the SIMP process was the identification of SIMP parameters.
The vertical slipper stiffness was defined as the deflection of the slipper beam at its
mid span when considering the most likely linear boundary conditions which are that
of a simply supported beam. This is justified since no significant deflection occurs
until both slippers are engaged with the rail and no moment support is reasonable for
these conditions since beam deflection will not cause the slipper to rotate through the
161
Table 7.10 SIMP Initial Design Parameters SIMP Design Parameter
Sled Mass
(lbf s2/in)
Length CG to Slipper
(in)
Pitching MMOI
(in lbf s2)
Yawing MMOI
(in lbf s2)
Vertical Slipper
Stiffness (lbf/in)
Lateral Slipper
Stiffness (lbf/in)
Front 2.181 59.94 4,708.14 4,692.14 860,732 146,332
Aft 2.181 60.73 4,708.14 4,692.14 860,732 146,332
Table 7.11 SIMP Final Design Parameters SIMP Design Parameter
Sled Mass
(lbf s2/in)
Length CG to Slipper
(in)
Pitching MMOI
(in lbf s2)
Yawing MMOI
(in lbf s2)
Vertical Slipper
Stiffness (lbf/in)
Lateral Slipper
Stiffness (lbf/in)
Front 2.022 59.94 4,653.78 4,639.34 5,82041 145,377
Aft 2.022 60.73 4,653.78 4,639.34 5,82041 145,377
slipper gap and bind. The lateral stiffness is the active stiffness of the slipper beam
that resists the lateral movement of the sled. In this case, it is the longitudinal
stiffness of half of the slipper beam and the stiffness of the outside vibration isolation
material. The equivalent stiffness value of the springs in series is defined as the
lateral slipper stiffness. The second step involved the computation of four effective
mass distributions utilizing Equation 7.3. These values are given in Table 7.12 and
were based upon a pinned-pinned 1st mode modal mass representation of the sled.
The third step was to calculate the associated effective frequency using Equation 7.4
162
and yielded the data in Table 7.13. The fourth step was the calculation of lift-to-
weight ratios which was 0 for the vertical value with the lateral case being held to the
lowest value of 1. Steps 5 and 6 involved the determination and use of a design
impact velocity using the downtrack velocity and effective impact velocity values
Y
AeffAL
Y
FeffFL
P
AeffAV
P
FeffFV
IMl
MM
IMl
MM
IMl
MM
IMl
MM
2
2
2
2
1
1
1
1
+=
+=
+=
+=
(7.3)
Table 7.12 SIMP Effective Mass Values Meff FV
(lbf s2/in)
Meff FL (lbf s2/in)
Meff AV (lbf s2/in)
Meff AL (lbf s2/in)
0.789 0.777 0.788 0.775
effAL
ALAL
effFL
FLFL
effAV
AVAV
effFV
FVFV
MKf
MK
f
MK
f
MK
f
π
π
π
π
21
21
21
21
=
=
=
=
(7.4)
163
from Step 3, Table 7.14. The downtrack velocity value for the present study was
9,500 fps. The original SIMP monorail data for the velocity cases of 8,000 fps and
10,000 fps, reported by Greenbaum, Garner, and Platus (1973), were used to
interpolate to the velocity under consideration. The resulting impact velocities were
utilized in Equation 7.5 to calculate the maximum slipper force and are given in Table
7.15. The calculation of the maximum slipper force was multiplied by a numeric
constant of 5 as prescribed by Greenbaum et al. (1973) to include a conservative
factor of 2.5 for a thrusting monorail sled. This methodology was shown to be overly
conservative (Mixon 2006). Since the current application was on a non-thrusting
narrow gauge and that Mixon (1999b, 1999c) successfully used a numeric factor of 2
Table 7.13 SIMP Effective Impact Frequency f FV (Hz)
fAV (Hz)
fFL (Hz)
fAL (Hz)
136.68 137.78 68.36 68.91
Table 7.14 SIMP Effective Impact Velocity v FV
(in/s)
vAV (in/s)
vFL (in/s)
vAL (in/s)
32.5 33.0 12.8 13.5
effALALALAL
effFLFLFLFL
effAVAVAVAV
effFVFVFVFV
MvfF
MvfFMvfF
MvfF
π
π
π
π
2
22
2
=
=
=
=
(7.5)
164
Table 7.15 SIMP Maximum Slipper Force FFV (lbf)
FAV (lbf)
FFL (lbf)
FAL (lbf)
22030 22190 4332 4533
on a thrusting narrow gauge pusher much stiffer than the current study, a numeric
factor of 2 was used in Equation 7.5. Studying Equation 7.5 it was apparent that there
was no coupling of the vertical and lateral forces other than mass distribution
relationships. The data from step 6 was used in step 7 to produce inertial loading
which could then be applied to either discrete or continuous mass systems. In the
current study, the data output from step 6 was sufficient to produce the shear and
moment diagrams in Figures 7.11-7.14. Substituting Equations 7.3 and 7.4 into 7.5
and assuming the relationship of impact velocity to impact frequency given in
Equation 7.6, it can be shown from a quantitative viewpoint that SIMP is essentially a
function of stiffness, K, effective mass, Meff, downtrack Velocity, Vd, lift-to-weight
ratio, L/W, and a constant as shown in Equation 7.7. The relationship in Equation 7.7
shows the dependence upon mass to be much less than that of λ and that SIMP is
more a function of Vd and L/W. Applying the stress analysis based upon the shear
and moment diagrams and from the iteration of structure variations through the SIMP
process, the final study dimension values were obtained and are shown in Table 7.16.
They represented a decrease in weight of 61.52 lbf from the initial design giving a
final sled weight of 780.48 lbf. The minimum MS of 0.00 was located at both slipper
beam mid-spans. During the design iteration process, it was observed that reducing
165
-15000
-10000
-5000
0
5000
10000
15000
0 0.2 0.4 0.6 0.8 1
Length fraction of L=26.04 (in)
Forc
e (lb
f)
VerticalLateral
Figure 7.11 SIMP Slipper Beam Shear Diagram
-160000
-140000
-120000
-100000
-80000
-60000
-40000
-20000
0
0 0.2 0.4 0.6 0.8 1
Length fraction of L=26.04 (in)
Mom
ent (
in lb
f)
VerticalLateral
Figure 7.12 SIMP Slipper Beam Moment Diagram
166
-25000-20000-15000-10000
-50000
500010000150002000025000
0 0.2 0.4 0.6 0.8 1
Length fraction of L=120.67 (in)
Forc
e (lb
f)
VerticalLateral
Figure 7.13 SIMP Sled Body Shear Diagram
-3000000
-2500000
-2000000
-1500000
-1000000
-500000
0
0 0.2 0.4 0.6 0.8 1
Length fraction of L=120.67 (in)
Mom
ent (
in lb
f)
VerticalLateral
Figure 7.14 SIMP Sled Body Moment Diagram
25.0
**constant ⎟⎟⎠
⎞⎜⎜⎝
⎛=
effD M
KWLVv (7.6)
where V is the slipper impact velocity
167
Vd is the downtrack velocity L/W is the Lift to Weight ratio K is the applicable stiffness Meff is the effective mass
constant* 25.075.0
WLVMKF Deff= (7.7)
where F is the general SIMP force
Table 7.16 SIMP Final Study Dimensions
Parameter Slipper Beam
Height (in)
Slipper Beam
Width (in)
Slipper Beam
Thickness (in)
Sled Body Radius
(in)
Sled Body Thickness
(in)
Value 3.5 3.5 0.35 8.06 1.21
weight in the slipper beam increased flexibility which when coupled with the
corresponding weight reduction served to reduce the vonMises stress level. The
relationship of slipper impact velocity to slipper impact frequency is a higher slipper
impact frequency corresponds to a higher impact velocity which linearly increases
SIMP loading as shown in Equation 7.5.
7.2.3 η Design Study
The η force design study consisted of entering the design parameters given in
Table 7.5 into the interpolation tool and applying the resulting loads to the sled. The
process of η force calculation involves the monitoring of interrelated design
parameters of mass, DP1, and sled natural frequencies, DP5, DP6, DP7. These specific
168
design parameters are affected by the variation of the study dimensions and are what
ultimately contributes to the variation of the η force. This is most efficiently
accomplished using a FEM software package or spreadsheet application as changes in
structural members result in changes in the associated mass and usually the natural
frequencies of the sled. There are situations where the 1st fundamental frequency of a
particular component is higher than the 1st fundamental frequency of the system.
Thus the associated frequency design parameter does not change with the mass of the
system. This phenomenon must be closely monitored with a FEM package to ensure
that all frequency content is not overlooked. The initial vertical and lateral loading
was 29,728 lbf and 21,948 lbf respectively. After iteration, the final design
parameters shown in Table 7.17 correspond to the final study dimensions shown in
Table 7.18. This yielded the shear and moment diagrams shown in Figures 7.15-7.18.
The final sled weight of 767.91 lbf shows a decrease in weight of 74.09 lbf from the
initial design. The minimum MS of 0.0 was located at the slipper beam midpoint.
7.2.4 Comparison of Design Study Results
The results of the sled design comparison appear in Tables 7.19-7.21 and
show that η produces a basic sled design with lower resultant dynamic loading and
correspondingly lower weight giving opportunity for better sled performance. Tables
7.19 and 7.20 summarize and compare the peak moment values for the three methods
found in Figures 7.8, 7.10, 7.12, 7.14, 7.16, and 7.18. The application of the
λ method was very direct and efficient as shown by Equation 7.2. The efficiency of
169
Table 7.17 η Final Design Parameters for Mock Sled Design Design
Parameter DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8
Variable W (lbf)
SG (in)
RRV (in)
RRL (in)
ωV (Hz)
ωL (Hz)
ωTOR (Hz)
V (fps)
Value 767.9 0.125 0.0119 0.0147 79.9 84.5 85.9 9,500
Table 7.18 η Final Study Dimensions
Parameter Slipper Beam
Height (in)
Slipper Beam
Width (in)
Slipper Beam
Thickness (in)
Sled Body
Radius (in)
Sled Body Thickness
(in)
Value 3.5 3.5 0.216 8.06 1.26
-10000-8000-6000-4000-2000
02000400060008000
10000
0 0.2 0.4 0.6 0.8 1
Length fraction of L=26.04 (in)
Forc
e (lb
f)
VerticalLateral
Figure 7.15 η Slipper Beam Shear Diagram
170
-120000
-100000
-80000
-60000
-40000
-20000
0
0 0.2 0.4 0.6 0.8 1
Length fraction of L=26.04 (in)
Mom
ent (
in lb
f)
VerticalLateral
Figure 7.16 η Slipper Beam Moment Diagram
-20000
-15000
-10000
-5000
0
5000
10000
15000
20000
0 0.2 0.4 0.6 0.8 1
Length fraction of L=120.67 (in)
Forc
e (lb
f)
VerticalLateral
Figure 7.17 η Sled Body Shear Diagram
171
-2000000-1800000-1600000-1400000-1200000-1000000
-800000-600000-400000-200000
0
0 0.2 0.4 0.6 0.8 1
Length fraction of L=120.67 (in)
Mom
ent (
in lb
f)
VerticalLateral
Figure 7.18 η Sled Body Moment Diagram
Table 7.19 Slipper Beam Peak Moment Values
Method Lateral (in lbf) Vertical (in lbf)
λ -126,656 -211,093
SIMP -28,203 -143,419
η -71,441 -97,766
Table 7.20 Sled Body Peak Moment Values
Method Lateral (in lbf) Vertical (in lbf)
λ -2,793,329 -3,861,933
SIMP -988,903 -2,623,842
η -1,577,358 -1,770,329
172
Table 7.21 Final Design Study Results Comparison
Method
Sled Weight
(lbf)
ΔWeight from Initial (lbf)
CG Force Vertical
(lbf)
CG Force Lateral
(lbf)
λ 847.75 +5.75 66,330 38,658
SIMP 780.48 -61.52 44,221 8,865
η 767.91 -74.09 29,728 21,948
use is overshadowed by the resulting over-prediction of loads when compared to
SIMP and η forces. The λ design study yielded a sled with a 1.0% increase in weight
over the initial value while SIMP and η yielded weight decreases of 7.3% and 8.8%
respectively. The load comparison is a better indicator of the differences in the
methods. With respect to η, λ over predicts 123.1% vertical and 76.1% lateral and
SIMP over predicts 48.8% vertical and under predicts 59.6% lateral. The preceding
study of λ, SIMP, and η shows that λ tends to over predict dynamic force with
increasing velocity. Ultimately infinite velocity would lead to infinite loading and
does not consider other design parameters. This over prediction yields conservative
designs with unnecessarily strong and heavy components that greatly reduce
performance in the hypersonic velocity realm of sled testing. The application of
SIMP is a lengthy seven-step process that is suited to automation in a spreadsheet
application. However the current method requires reading impact velocity and
frequency charts which are only available as SLEDYNE generated output tables. The
inclusion of rail roughness and slipper gap is available from SLEDYNE through its
173
relationship of these values to slipper impact frequency and velocity. Unfortunately
SLEDYNE is practically unusable in its current state with the inability to produce
current slipper impact frequency and velocity charts making it a method with fixed
parameters with regard to rail properties. It should be noted that the slipper impact
frequency and velocity tables used in this study were originally developed for a
monorail sled since no comprehensive narrow gauge study had been completed at the
time of their production. Typically, monorail sleds have a worse vibration
environment than narrow gauge sleds. Also, in every study of dynamic sled loading
the downtrack sled velocity is a significant and often the dominant parameter in
dynamic load prediction so the use of charts specific to particular downtrack
velocities makes SIMP somewhat cumbersome in its use. It should be noted that
there is a limit within SIMP where dynamic loading goes to 0 as impact frequency
approaches 0. This aspect if considered alone could yield a sled component with
unrealistic flexibility. The fact remains that for realistic design parameters SIMP
produces loads in the vertical direction are realistic and suited to help produce
optimized designs. In the lateral direction SIMP under predicts loads by nearly 60%
compared to η. The η formation is more efficient than SIMP since it uses a FEM tool
to produce the natural frequencies that the analyst does by hand with SIMP. Both
methods give the analyst some latitude in choosing significant frequency components
and rely on an in-depth knowledge of the mass distribution and stiffness of the sled
system, where λ uses only mass and velocity. Both SIMP and η are found to deviate
from the lateral to vertical ratio of 0.6 as documented by Krupovage et al. (1991).
174
Iteration through the η method showed that dynamic loading was dependent upon all
structural aspects of the sled including mass and stiffness distribution. SIMP in this
application is constrained to the flexibility of the slipper beam for its frequency
content, and thus any variations in the sled body do not affect SIMP loading. λ
produced loading is entirely deficient in taking into account sled body flexibility.
7.3 CO Test Data Comparison
A comparison of η force to corresponding CO test data resolved from slipper
beam force at the sled CG was conducted to evaluate the effectiveness of η in
predicting dynamic load. Of all the tests studied only the CO test returned data
suitable for resolving at the sled CG in the vertical direction. Figure 7.19 details the
application of the η for typical values of all design parameters except mass/weight
which for this case was 770 lbf. The velocity region shown is represents maximum
velocity of the CO test. η force successfully bounds the force data in all but three
areas. These areas are associated with probable noise spikes from the original data
trace.
175
-40000
-30000
-20000
-10000
0
10000
20000
30000
40000
4800 4900 5000 5100 5200 5300 5400 5500
Velocity (fps)
Forc
e (lb
f)
CO Test Vertical CG ForceEta 222222X
Figure 7.19 CO Test Vertical Force Comparison
176
8. DISCUSSION OF RESULTS
Several results from the execution of this study were deemed significant and
worthy of documentation. There were original results generated that contributed to
the advancement of narrow gauge sled design in the area of dynamic modeling and
load prediction since lateral slipper rail contact and resulting force data were studied
in detail for the first time. Further, this study represents the first large scale numerical
study of a narrow gauge sled system.
8.1 Major Findings
Several major findings related to the modeling and prediction of dynamic
loading of narrow gauge sleds were revealed in this study.
8.1.1 λ and SIMP Limits of Applicability
An initial review of the literature and examination of previous loading
methods showed that λ and SIMP had narrow limits of applicability. λ was found to
be applicable to similar test data points from which it was derived and SIMP was
found to be applicable to vertical loading. There were several instances from actual
test data that illustrated how λ was not accurate in predicting narrow gauge dynamic
loading. In most instances, λ over predicted loading but, for a few cases under
predicted loading when compared to test data (Mixon and Hooser 2002).
Additionally, both methods, SIMP and λ, were found to over predict loading for the
177
all-typical case, 2222222, when compared to η at higher velocities. Outside of their
narrow range of applicability, both methods return erroneous results.
8.1.2 Applicability of Dynamic Structural Modeling Approaches
Suitable structural modeling assumptions were discovered for direct
representation of narrow gauge sleds in DADS for the purposes of successful
transient dynamic modeling. Two investigations were conducted into the
implementation of modal and SMD representations of structural components. The
SMD representation employed Euler-Bernoulli beam theory to represent the structural
flexibility between discretized masses. The modal representation utilized mode
shapes and natural frequencies built using Timoshenko beam theory to define
structural flexibility. For linear boundary conditions the modal representation was
found to be acceptable while the SMD system was acceptable regardless of boundary
conditions and was subsequently used in correlation modeling. The robustness of the
SMD representation was well-suited for the nonlinear hypervelocity environment
experienced during simulated tests of the narrow gauge sled in this study. Correlation
modeling showed that the SMD representation also quickly correlated in areas near
the slipper rail interface better representing the dynamic load generation of the
slipper-rail interface.
8.1.3 Slipper Rail Impact Characterization
Slipper rail impact for the slipper configuration particular to a narrow gauge
178
sled was found to be the primary basis of the dynamic load generation in the sled.
From a nonlinear FEM study, slipper rail contact values were returned for three
distinct impact configurations; and for the first time, lateral contact characterization
was described. It showed a significant amount of rail head torsional displacement
which absorbs considerable amounts of linear slipper momentum. During the course
of correlation modeling, it was not until suitable contact stiffness and damping values
were deduced and implemented into the entire sled model that the correlation
numbers reached acceptable values. This was evidenced by the high correlation
values, average FOMs of 75.6 frequency and 85.2 standard deviation, in the vicinity
of the slipper beam and rocket motor near the slipper beam attachments.
8.1.4 Importance of Modeling of Sled Nonlinearities
Nonlinearities in modeling such as nonlinear stiffness and contact were
critical to the production of a correlated model. Sled motion and subsequent loading
is heavily influenced by the nonlinear contact scenario generated by the movement of
the sled through the slipper gap during a test. During the course of correlation
modeling it was discovered that sled movement and loading was also influenced by
the accurate modeling of the nonlinearities elsewhere on the sled. The extent of
accurate correlation modeling was dictated locally by the analyst’s ability to account
for these other nonlinear features. The following nonlinear features were found to be
influential: slotted contact between aft payload and rocket motor and nonlinear
stiffness in the payload and sled vibration isolation systems.
179
8.1.5 Importance of Accurate Damping Modeling
During the course of developing the correlation model, the values of damping
associated with the payload and sled vibration isolation systems were found to be
important. Prescribing an artificially high damping value leads to an under prediction
of acceleration and the generation of artificially high forces at the area under
consideration, as well as throughout the entire sled system. It also rendered realistic
stiffness values irrelevant as the damping mechanism dominated the force generation.
Hypersonic sled systems are particularly sensitive to excessive damping values as
high velocity values abound and the problem can quickly escalate. It was also seen
that excessive damping for some isolated instances appeared to be suitable; however,
when other modeling parameters were varied slightly the excessive damping
produced erroneous results.
8.1.6 Utility of Figure of Merit (FOM)
During the course of correlation modeling it was found that that the most
efficient way to compare numerical data to test data was to use FOM comparisons at
test measurement locations. The use of weighted averages in the frequency domain
was particularly useful since convergence was more meaningful as individual FOM
values increased. The use of standard deviation as the time domain value was well
suited to the current study since peak force prediction was the ultimate goal of the
study lending itself to the direct measure of model validity in this area.
180
8.1.7 Identification of Appropriate Design Parameters and Corresponding Ranges
Design parameter variation of a reduced complexity model was done to address
generic sled designs of the immediate future at the HHSTT by choosing typical
values associated with an LSR forebody sled with the payload removed and then by
varying the associated parameters through a realistic expected range. It was found
that the range employed was acceptable and also that some prudence in choosing the
number and value of the design parameters is warranted since the resulting number of
possible simulations is quite large and time consuming to complete. The total number
of possible simulations in the current study was 2187 where on a five processor
system the average run time of one simulation was 9.1 hours making the solution of
all simulations roughly 166 days. Selectively choosing a reduced set of 303
simulations, representing the extremum η force values, reduced the solution time to
23 days.
8.1.8 Superiority of Multivariate Interpolation to Correlate Design Parameters to η Forces
Study of the parameter variation solutions indicated that a basic multivariate
least squares correlation produces reasonable correlation but not in sufficient detail
for a designer to accurately predict dynamic loading. This is due to the nonlinear
nature of the dynamic load data and from the piecewise linear nature with which the
data were reported as it varied over velocity. The piecewise linear nature is from the
enveloping of the data to afford some conservatism in the formation of η force since
181
it bypassed certain velocity ranges where the dynamic load locally decreased. The
most accurate representation of η force resulted from multivariate interpolation.
8.1.9 Influence of Design Parameters on Dynamic Loading
Velocity was found to be the most influential design parameter in the
magnitude of dynamic load since increases in velocity always cause an increase in
dynamic load. This is a natural conclusion and supported by other studies of this
nature (ISTRACON 1961, Mixon 1971, Hooser and Mixon 2002) and experimental
work (Hooser 1989, Leadingham and Schauer 2001, Turnbull and Minto 2003). It
was also found that rail roughness (vertical and lateral), slipper gap, and translational
sled natural frequency were the main contributing factors in the dynamic loads. η
force values when compared to corresponding values from λ and SIMP methods were
significantly lower for both force directions except for some instances of lateral
SIMP. After velocity, rail roughness was shown to be the second most influential
design parameter since the high value of rail roughness is extreme, 30% higher than
typical or surveyed data.
8.2 Significance and Contribution of Current Study
The significance of this study is the advancement of the modeling of
hypersonic narrow gauge sleds, specifically in the representation of the structural
members and characterization of slipper rail contact. The method of structural
modeling was unique since it was the first attempt to use a method directly and
182
entirely related to the structural geometry and material of the particular component
and was able to bypass, in all cases but one, laboratory testing to verify the model.
There was significant information produced that verified the proper modeling of
slipper rail contact in the vertical direction which directly lead to a greater assurance
that characterization of the lateral impact scenario was also properly modeled. These
modeling efforts were verified by the first successful comprehensive correlation of a
hypersonic narrow gauge sled using an expanded form of the correlation technique
introduced by Hooser and Schwing (2000).
The second area of contribution was in the area of dynamic load prediction.
The general method was first used by Mixon (1971) and involved the use of a verified
model to numerically expand the input parameters to extrapolate a numerical model
to produce dynamic load predictions. The current study applied this technique to a
narrow gauge sled system in a hypervelocity environment and produced vertical and
lateral force data in the form of η forces. This is the first known narrow gauge sled
dynamic load prediction tool developed from in depth correlation studies of such a
system.
183
9. CONCLUSIONS
The current study tested and selected two methods of structural representation,
SMD and modal, and developed and implemented a new slipper rail contact scheme
to build numerical models of hypervelocity narrow gauge sleds. These models were
developed to the point that an acceptable correlation, based on frequency and
standard deviation FOM numbers, was returned. The modeling method and
corresponding assumptions were applied to a reduced complexity narrow gauge sled
model where eight critical design parameters were identified. The eight design
parameters: mass, slipper gap, rail roughness vertical, rail roughness lateral, sled
natural frequency vertical, sled natural frequency lateral, sled natural frequency
torsional, and velocity, were varied through strategic combinations of their
corresponding high, low, and typical values. The resulting dynamic load data, η
force, was studied and compared to previous load prediction methods, λ and SIMP.
For direct comparisons, η force was found to be generally lower than previous
methods and resulted in a more efficient basic sled design in terms of total sled
weight.
9.1 Existing Theory Review of Dynamic Load Prediction
The review of existing theory indicated that closed form dynamic load
prediction had been attempted λ (ISTRACON Handbook 1961) and SIMP
(Greenbaum, Garner, Platus 1973, Mixon 1971). Some success was met with λ
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becoming the standard at the HHSTT for monorail, narrow gauge, and dual rail sleds.
λ was shown to be typically conservative but for certain conditions, it under predicted
dynamic loading (Mixon and Hooser 2002). The principal deficiencies of SIMP are
its non-correlation in the lateral direction, and its inability to produce slipper impact
velocities and frequencies indicative of varied slipper gaps and rail roughness.
9.2 Numerical Model Development
Numerical modeling development was accomplished by creating two distinct
DADS sled models based on different structural representations. Slipper rail impact
was developed in PRONTO, studied, and implemented in both DADS models.
Connections between major structural components were implemented in the DADS
models to complete the numerical modeling process.
9.2.1 Modeling Assumptions
For the current study the following items were assumed to exist: elastic slipper
rail contact, flexible major structural components, rough rail, and rigid rail. The
following items were assumed to either be nonexistent or to be not critical to
numerical sled modeling in this effort: plastic deformation at slipper rail contact,
slipper gouging, significant aerodynamic or frictional heating, frictional slipper wear,
and a flexible rail. This selection of modeling assumptions was intended to capture
the majority effect of relevant structural dynamics that characterize a rocket sled
traversing the rail.
185
9.2.2 Slipper Rail Impact Results
Finite element modeling of the slipper and rail in PRONTO, a nonlinear FEM
package, was accomplished to help uncover the true nature of slipper rail impact. The
study centered on approaching a flexible rail from three directions typical to slipper
rail impact: vertical down, lateral, and vertical up at typical impact velocities: 10 ips,
25 ips, 50 ips, and 100 ips. The downtrack velocity of the slipper was 5,000 fps, the
approximate velocity midpoint of the LSR test. The results of the vertical impacts
resulted in the vertical up and down directions losing slightly more translational
momentum, COR of 0.5 and 0.4 respectively, than the classic sphere on plate COR of
0.7. The lateral impact case showed significant rail head torsional motion which
allowed the slipper to expend roughly twice the translational momentum of the
vertical impacts, COR of 0.2. The resulting slipper rail contact parameters were
implemented in DADS for a rigid rail and the corresponding results correlated to
those in PRONTO.
9.2.3 DADS Implementation
Both structural representations were implemented in DADS in addition to the
contact characterization derived from the PRONTO results. Stiffness and damping
values associated with structural connections of the major structural components were
prescribed along with a rough rigid rail described by survey data. Appropriate lift,
drag, and thrust values were applied so that the sled simulated the LSR and CO tests.
186
9.3 Evaluation of Numerical Model
The numerical model was evaluated to gauge its accuracy against measured
data sets from the LSR and CO tests so that the underlying modeling assumptions
could be verified. The validation of the numerical, DADS generated, data to the test
data was accomplished through the use of frequency and time domain comparisons.
The data was band pass filtered from 10 Hz to 4500 Hz and then converted to a
velocity windowed data set and the standard deviation over the velocity window was
computed every 250 fps to constitute the time domain comparison. The standard
deviation values were compared to the curve fit of the correspondingly processed test
data and the RMS weighted average deviation was returned as the time domain FOM.
The frequency domain comparison consisted of band pass filtering and then
converting the data to the frequency domain using an FFT algorithm and then
integrating the data to evaluate the RMS power of the data at intervals of every 50
Hz. The deviation of the DADS data from the correspondingly processed test data
was weighted using the test data ratio of RMS power at each frequency interval and
the RMS deviation was returned to form the frequency FOM. FOM values for eleven
data channels were averaged and used in the correlation of the DADS model to the
test data. Correlation criteria were that the average frequency FOM was greater than
70 with no individual value less than 50 and the average time domain FOM was
greater than 80 and no individual value less than 70. The SMD structural
representation was found to most readily meet the criteria with average FOM values
of 60.08 frequency, 85.79 STD, and 71.06 frequency, 87.80 STD for the LSR and CO
187
tests respectively. The exception given to the SMD frequency FOM for the LSR test
is due to the in ability to characterize rocket motor combustion instability and from
the excellent FOM numbers near the slipper rail interface.
9.4 Evaluation of Large Scale Design Parameter Variation
The modeling assumptions validated by evaluating the numerical model were
implemented in a reduced complexity model indicative of a LSR forebody sled with
the payload removed and the fuel half expended. Eight design parameters were
identified that represented the most critical dynamic load contributors. The design
parameters were then assigned typical, low, and high values indicative of reasonable
variations of each. The design parameters associated with slipper gap and rail
roughness were assigned typical values as surveyed or prescribed by HHSTT policy.
The high and low values of slipper gap were prescribed by limits found in HHSTT
policy where the high and low limits of rail roughness were prescribed by a ± 30%
variation. The design parameters associated with the mass of the sled were varied to
represent the rocket motor of the LSR forebody sled at full burnout and at motor
ignition. The velocity of the sled was varied from rest to 10,000 fps. The remaining
design parameters associated with sled natural frequency were evaluated at values
typical to the half expended LSR forebody and then at values representing a high and
low of ± 50% respectively. Velocity was the independent variable over which force
data was evaluated so simulations involving design parameters involved the
monitoring of only seven design parameters. The design parameters were varied and
188
simulations run through cycled combinations to return η force data in the vertical and
lateral directions. The initial design parameter variation consisted of the all typical,
2222222, case where initial force values were evaluated. The next variation included
varying the design parameters by setting only one design parameter to either extreme,
1 or 3, and keeping all others at the typical value of 2. This variation was
accomplished to determine the effect of the design parameters as individual values
were cycled through their extreme values. The next variation consisted of all possible
combinations of the high and low values, 1 and 3, 128 simulations total. This
variation was accomplished to establish an extremum set defining loading at the
design parameter bounding values. The final design parameter variation consisted of
varying the design parameters through a reduced set of design parameter values
where noncontributing design parameter values were excluded. This variation was
accomplished to round out the study by investigating any possible excessive dynamic
loads without completing the overbearing number of simulations, 2187, possible with
the variation of all parameters involved 37. This final set initially numbered 192 but
since 32 simulations had been previously completed the total number of simulations
was reduced to160.
9.5 Correlation of Predicted Load and Design Parameters
The correlation of the numerically generated force data to the design
parameters was begun by forming η forces from the peak values of the DADS data.
η force is an envelope function that bounds the peaks of the data as plotted over
189
velocity causing η to be piecewise linear over velocity and the other design
parameters. The correlation of η was initiated using linear regression to investigate
the possibility of a closed form solution to relate the design parameters to the vertical
and lateral η forces. This attempt was not successful and the implementation of a
multivariate interpolation technique was employed to relate design parameter values
to output η force. The use of a multivariate interpolator does not immediately make
the relationship of the design parameters to the η forces apparent but does guarantee
that the values of calculated η force are an accurate representation of the DADS data
and makes η useful to design engineers.
9.6 Comparison to Previous Methods
The η force values were compared to the corresponding values calculated
from the λ and SIMP methods. It was found that η returned lower values of dynamic
load for velocities above the force transition point typically between 800 fps and
2,000 fps. For the all typical case, 2222222, η was shown to be significantly lower
than λ at high velocities and consistently lower than SIMP for all velocities in the
vertical direction and under 4,000 fps in the lateral direction. The lower values of
SIMP were due to SIMP not being correlated in the lateral direction due to inadequate
slipper rail impact characterization. It was shown that η was able to account for the
changes in rail roughness, while λ and SIMP utilize fixed typical values of rail
roughness and slipper gap. As expected, η force values increased with a large rail
roughness increase of 30%. A basic sled design which consisted of a sled body and
190
slipper beams was completed for each method. The basic sled design was analyzed
and redesigned based upon each of the three methods. It was seen that the λ−based
design yielded a mass increase of 1% over the baseline, the SIMP-based design
resulted in a mass decrease of 7.3% and the η-based design resulted in a mass
decrease of 8.8%. When compared to η, λ over predicts design loads by 123.1%
vertical and 76.1% lateral; SIMP over predicts loads by 48.8% vertical and under
predicts by 59.6%. Again the reason for the under prediction of SIMP is from
inaccurate characterization of the lateral slipper rail impact.
9.7 Future Work
There is great opportunity to pursue further work in the general area of
hypersonic sled dynamic load prediction and into many other related areas.
Recommendations for further study include expanding the formation of η force
calculation into the areas of monorail and dual rail sled systems. Since there have
been many more sled tests and consequently more and various types of sleds
constructed for these two configurations than for narrow gauge systems, the breadth
and depth of study and correlation would lead to a comprehensive dynamic load
prediction. Monorail and dual rail sled systems represent a marked departure in
design philosophy from each other and narrow gauge sled systems; however, the
proper application of the dynamic load prediction method outlined in this work would
return useful data from which a suitable load prediction database could be
constructed. Narrow gauge results show λ and SIMP to be inadequate in accurately
191
predicting dynamic loading over the wide range of design parameters encountered
during the range of HHSTT test applications. Further work into the area of narrow
gauge dynamic load prediction would include analyzing narrow gauge sleds that
deviate from the typical slipper beam and sled body design, where the more rigid
truss structure type sleds are studied. This type of study would effectively build upon
the established η force narrow gauge database.
Structural modeling may be optimized by studying discretization limits of a
robust beam theory such as the Timoshenko theory to reduce the number of
discretized masses necessary to construct an accurate DADS sled model. This was
accomplished to some extent in the current study but it is likely that further reductions
may be realized by detailing the lower limit of discretized masses needed in DADS.
Further studies into the nature of the slipper rail impact for different structural
representations of slippers in general would uncover the driving mechanism behind
slipper impact and could eventually lead to improved slipper isolation mechanisms.
It could also lead to engineered slippers with increased flexibility that would help
reduce dynamic loading through momentum exchange with the rail.
Additional work into the correlation of the input design parameters to output η
forces can be accomplished where the nonlinear nature of the output data may be
captured by nonlinear methods into a closed form. A successful closed form outcome
is doubtful but its completion would be valuable enough to sled designers and load
analysts alike that the required work would be warranted. A closed form solution
built from accurate loading data would immediately make dynamic load
192
characterization apparent and useful.
9.8 Concluding Remarks
The results of this study have shown that accurate modeling and
characterization of a narrow gauge sled was accomplished and that comprehensive
correlation to test data was achieved. Numerical modeling based upon this
correlation effort was accomplished to produce a new method of predicting dynamic
loading in the lateral and vertical directions based upon high fidelity numerical sled
tests. The resulting data were correlated in a form similar to λ in ease of use while
retaining the capability to return accurate results. This effort gives sled designers a
new capability to confidently produce optimized narrow gauge sled designs.
193
APPENDIX A
FEMS AND RESULTING COMPARISONS
194
Figure A.1 Rocket Motor Beam FEM
195
Figure A.2 Slipper Beam Three Dimensional FEM
196
Figure A.3 Slipper Three Dimensional FEM
197
Figure A.4 Payload Three Dimensional FEM
198
Table A.1 Modal Model Payload Mass Properties Comparison
Model Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
3D Solid 145.43 194.00 194.00 7.78
DADS/Flex 141.14 185.1 184.9 5.71
Deviation % -2.9 -4.5 -4.7 26.6
Table A.2 Modal Model Payload Frequency Comparison
Model Boundary Condition Frequency (Hz) and Mode Shape
3D FEM
Fixed (FWD) –
Free (AFT)
151 1st B V*
152 1st
B L*
803 2nd
B V*
833 2nd
B L*
3500 1st
DT*
DADS/Flex
Fixed (FWD) –
Fixed (AFT)
134 1st B V*
160 1st
B L*
767 2nd
B V*
833 2nd
B L*
3487 1st
DT*
Deviation % -11.2 5.3 -4.5 0.0 -0.4
* V-Vertical; L-Lateral; DT-DownTrack; B-Bending; T-Torsional
199
Table A.3 Modal Model Slipper Beam Mass Properties Comparison
Model Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
3D Solid 49.68 1.29 5.29 5.85
DADS/Flex 49.91 1.17 5.30 5.89
Deviation % 0.5 -9.3 0.2 0.7
Table A.4 Modal Model Slipper Beam Frequency Comparison
Model Boundary Condition Frequency (Hz) and Mode Shape
3D FEM
Fixed at Slipper
Beam Center
259 SS V*
487 SS DT*
650 C DT*
732 1st C V*
1850 1st T*
DADS/Flex
Fixed at Slipper
Beam Center
273 SS V*
566 SS DT*
799 C DT*
682 1st C V*
1512 1st T*
Deviation % 5.5 16.2 22.9 -6.8 -18.3
* V-Vertical; L-Lateral; DT-DownTrack; SS-See Saw; T-Torsional; C-Cantilever
200
Table A.5 Modal Model Slipper Mass Properties Comparison
Model Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
3D Solid 27.06 0.803 0.770 0.521
DADS/Flex 25.99 0.616 0.595 0.504
Deviation % 4.0 -23.3 -22.7 -3.3
Table A.6 Modal Model Slipper Frequency Comparison
Model Boundary Condition Frequency (Hz) and Mode Shape
3D FEM Fixed at SB
interface
708 B L*
2280 B V*
DADS/Flex
Fixed (FWD) –
Fixed (AFT)
770 B L*
2065 B V*
Deviation % 8.8 -9.4
* V-Vertical; L-Lateral; B-Bending
201
Table A.7 Spring Mass Damper Model Payload Mass Properties Comparison
Model Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
3D Solid 145.25 194.00 194.00 7.79
DADS/SMD 145.25 205.70 205.70 7.87
Deviation % 0.0 6.0 6.0 1.0
Table A.8 Spring Mass Damper Model Payload Frequency Comparison
Model Boundary Condition Frequency (Hz) and Mode Shape
3D FEM
Fixed (FWD) –
Fixed (AFT)
151 1st B V*
152 1st
B L*
803 2nd
B V*
833 2nd
B L*
3500 1st
DT*
DADS/SMD
Fixed (FWD) –
Fixed (AFT)
158 1st B V*
141 1st
B L*
803 2nd
B V*
660 2nd
B L*
3400 1st
DT*
Deviation % 4.6 -7.2 0.0 -20.7 -2.9
* V-Vertical; L-Lateral; DT-DownTrack; B-Bending; T-Torsional
202
Table A.9 Spring Mass Damper Model Slipper Beam Mass Properties Comparison
Model Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
3D Solid (no Yoke) 32.23 0.34 4.55 4.54
DADS/SMD 32.23 0.34 4.76 4.76
Deviation % 0.0 0.0 4.6 4.8
Table A.10 Spring Mass Damper Model Slipper Beam Frequency Comparison
Model Boundary Condition Frequency (Hz) and Mode Shape
3D FEM (no yoke)
Fixed at Slipper
Beam Center
279 SS V*
578 SS DT*
697 1st C DT*
750 1st C V*
1900 1st T*
DADS/SMD
Fixed at Slipper
Beam Center
270 SS V*
580 SS DT*
650 1st C DT*
780 1st C V*
2700 1st T*
Deviation % 3.2 0.3 -6.7 4.0 42.0
* V-Vertical; L-Lateral; DT-DownTrack; SS-See Saw; T-Torsional; C-Cantilever
203
Table A.11 Spring Mass Damper Model Slipper Mass Properties Comparison
Model Weight (lbf)
Ixx (lbf s2 in)
Iyy (lbf s2 in)
Izz (lbf s2 in)
3D Solid 27.06 0.803 0.770 0.521
DADS/SMD 26.93 0.759 0.697 0.482
Deviation % -0.5 -5.5 -9.5 -7.5
Table A.12 Spring Mass Damper Model Slipper Frequency Comparison
Model Boundary Condition Frequency (Hz) and Mode Shape
3D FEM Fixed at SB
interface
708 B L*
2280 B V*
DADS/Flex
Fixed (FWD) –
Fixed (AFT)
683 B L*
2080 B V*
Deviation % -3.5 -8.8
* V-Vertical; L-Lateral; B-Bending
204
APPENDIX B
FLEXIBLE BODY STIFFNESS FORMULATION
205
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APPENDIX C
MATLAB CORRELATION DATA PROCESSING SCRIPTS
209
data_master.m Main Program File
% data master % this script calls both processing scripts for DADS % and mission data and then calls the plotting script % Mission data processing, doesn't need to be run all % of the time sw_md=input('Process mission data? '); if sw_md, data_proc, end % DADS data processing sw_dd=input('Process DADS data? '); if sw_dd, data_proc_dads, end % Plot this data pp=input('Plot this data? '); if pp, data_plot, end
Published with MATLAB® 7.0
210
data_proc.m Test Data Processing Script
% This script is designed to process data from the % 80X-G1 mission by filtering, then rearranging the % data into velocity windowed format and performing % PSD analysis. Finally the data is written into % *.txt files This script does not perform any data % decimation % S Furlow clear all % Load raw data from file data=load('D80XG1F_no_head.txt'); data_end=60394; % Assign data more meaningful names d_141=data(10001:80001,13); d_140=data(10001:80001,12); d_139=data(10001:80001,11); d_138=data(10001:80001,10); d_137=data(10001:80001,9); d_124=data(10001:80001,8); d_123=data(10001:80001,7); d_115=data(10001:80001,6); d_113=data(10001:80001,5); d_112=data(10001:80001,4); d_103=data(10001:80001,3)-8.5; d_101=data(10001:80001,2); t=data(10001:80001,1); % Load velocity dum=load('80x_g1_data.txt'); vel=dum(:,2); vel_size=numel(vel) if vel_size<70001
for i=vel_size+1:70001, vel(i)=vel(vel_size)+100; end
end % Filter Data
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dt=t(2)-t(1); fs=1/dt; nyq=fs/2; wn=[10/nyq,4500/nyq]; [B,A] = BUTTER(3,wn); d_101_f = filter(B,A,d_101); d_103_f = filter(B,A,d_103); d_112_f = filter(B,A,d_112); d_113_f = filter(B,A,d_113); d_115_f = filter(B,A,d_115); d_123_f = filter(B,A,d_123); d_124_f = filter(B,A,d_124); d_138_f = filter(B,A,d_138); d_139_f = filter(B,A,d_139); d_140_f = filter(B,A,d_140); d_141_f = filter(B,A,d_141); % Build Velocity Windowed data d_vel=250; j=1; vel_i=vel(1); for i=1:numel(vel),
if (vel(i)-vel_i)>d_vel, index(j)=i; j=j+1; vel_i=vel(i); end
end index_vel_end=j; for i=1:index_vel_end-2,
vel_a(i)=mean(vel(index(i):index(i+1))); d_101_f_std(i)=std(d_101_f(index(i):index(i+1))); d_103_f_std(i)=std(d_103_f(index(i):index(i+1))); d_112_f_std(i)=std(d_112_f(index(i):index(i+1))); d_113_f_std(i)=std(d_113_f(index(i):index(i+1))); d_115_f_std(i)=std(d_115_f(index(i):index(i+1))); d_123_f_std(i)=std(d_123_f(index(i):index(i+1))); d_124_f_std(i)=std(d_124_f(index(i):index(i+1))); d_138_f_std(i)=std(d_138_f(index(i):index(i+1))); d_139_f_std(i)=std(d_139_f(index(i):index(i+1))); d_140_f_std(i)=std(d_140_f(index(i):index(i+1))); d_141_f_std(i)=std(d_141_f(index(i):index(i+1)));
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dum_101=sort(abs(d_101_f(index(i):index(i+1))),'descend'); dum_103=sort(abs(d_103_f(index(i):index(i+1))),'descend'); dum_112=sort(abs(d_112_f(index(i):index(i+1))),'descend'); dum_113=sort(abs(d_113_f(index(i):index(i+1))),'descend'); dum_115=sort(abs(d_115_f(index(i):index(i+1))),'descend'); dum_123=sort(abs(d_123_f(index(i):index(i+1))),'descend'); dum_124=sort(abs(d_124_f(index(i):index(i+1))),'descend'); dum_138=sort(abs(d_138_f(index(i):index(i+1))),'descend'); dum_139=sort(abs(d_139_f(index(i):index(i+1))),'descend'); dum_140=sort(abs(d_140_f(index(i):index(i+1))),'descend'); dum_141=sort(abs(d_141_f(index(i):index(i+1))),'descend'); d_101_f_p1(i)=dum_101(1); d_103_f_p1(i)=dum_103(1); d_112_f_p1(i)=dum_112(1); d_113_f_p1(i)=dum_113(1); d_115_f_p1(i)=dum_115(1); d_123_f_p1(i)=dum_123(1); d_124_f_p1(i)=dum_124(1); d_138_f_p1(i)=dum_138(1); d_139_f_p1(i)=dum_139(1); d_140_f_p1(i)=dum_140(1); d_141_f_p1(i)=dum_141(1); d_101_f_p2(i)=dum_101(2); d_103_f_p2(i)=dum_103(2); d_112_f_p2(i)=dum_112(2); d_113_f_p2(i)=dum_113(2); d_115_f_p2(i)=dum_115(2); d_123_f_p2(i)=dum_123(2); d_124_f_p2(i)=dum_124(2); d_138_f_p2(i)=dum_138(2); d_139_f_p2(i)=dum_139(2);
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d_140_f_p2(i)=dum_140(2); d_141_f_p2(i)=dum_141(2);
end % Frequency analysis t_off=43751; [P_101_f,F_101] = PWELCH(d_101_f(1+t_off:8600+t_off),[],[],[],fs); [P_103_f,F_103] = PWELCH(d_103_f(1+t_off:14445+t_off),[],[],[],fs); [P_112_f,F_112] = PWELCH(d_112_f(1+t_off:15099+t_off),[],[],[],fs); [P_113_f,F_113] = PWELCH(d_113_f(1+t_off:15099+t_off),[],[],[],fs); [P_115_f,F_115] = PWELCH(d_115_f(1+t_off:14445+t_off),[],[],[],fs); [P_123_f,F_123] = PWELCH(d_123_f(1+t_off:15099+t_off),[],[],[],fs); [P_124_f,F_124] = PWELCH(d_124_f(1+t_off:15099+t_off),[],[],[],fs); [P_138_f,F_138] = PWELCH(d_138_f(1+t_off:14445+t_off),[],[],[],fs); [P_139_f,F_139] = PWELCH(d_139_f(1+t_off:15099+t_off),[],[],[],fs); [P_140_f,F_140] = PWELCH(d_140_f(1+t_off:15099+t_off),[],[],[],fs); [P_141_f,F_141] = PWELCH(d_141_f(1+t_off:15099+t_off),[],[],[],fs); % Export data fid =fopen('d_101_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_101,P_101_f]'); fid =fopen('d_103_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_103,P_103_f]'); fid =fopen('d_112_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_112,P_112_f]'); fid =fopen('d_113_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_113,P_113_f]'); fid =fopen('d_115_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_115,P_115_f]'); fid =fopen('d_123_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_123,P_123_f]'); fid =fopen('d_124_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_124,P_124_f]'); fid =fopen('d_138_psd.txt','w');
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fprintf(fid,'%6.4f %12.6f\n',[F_138,P_138_f]'); fid =fopen('d_139_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_139,P_139_f]'); fid =fopen('d_140_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_140,P_140_f]'); fid =fopen('d_141_psd.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_141,P_141_f]'); fid =fopen('d_101_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_101_f_std',d_101_f_p1',d_101_f_p2']'); fid =fopen('d_103_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_103_f_std',d_103_f_p1',d_103_f_p2']'); fid =fopen('d_112_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_112_f_std',d_112_f_p1',d_112_f_p2']'); fid =fopen('d_113_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_113_f_std',d_113_f_p1',d_113_f_p2']'); fid =fopen('d_115_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_115_f_std',d_115_f_p1',d_115_f_p2']'); fid =fopen('d_123_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_123_f_std',d_123_f_p1',d_123_f_p2']'); fid =fopen('d_124_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_124_f_std',d_124_f_p1',d_124_f_p2']'); fid =fopen('d_138_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_138_f_std',d_138_f_p1',d_138_f_p2']'); fid =fopen('d_139_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_139_f_std',d_139_f_p1',d_139_f_p2']'); fid =fopen('d_140_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_140_f_std',d_140_f_p1',d_140_f_p2']'); fid =fopen('d_141_f_v.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n', [vel_a',d_141_f_std',d_141_f_p1',d_141_f_p2']'); fid = fopen('d_101_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_101_f]'); fid = fopen('d_103_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_103_f]');
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fid = fopen('d_112_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_112_f]'); fid = fopen('d_113_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_113_f]'); fid = fopen('d_115_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_115_f]'); fid = fopen('d_123_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_123_f]'); fid = fopen('d_124_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_124_f]'); fid = fopen('d_138_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_138_f]'); fid = fopen('d_139_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_139_f]'); fid = fopen('d_140_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_140_f]'); fid = fopen('d_141_f.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_141_f]'); fclose('all')
Published with MATLAB® 7.0
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data_proc_dads.m DADS Data processing script
% This script is designed to process DADS generated % data of the 80X-G1 mission by filtering, then % rearranging the data into velocity windowed % format and performing PSD analysis. Finally the % data is written into *.txt files This script does % not perform any data decimation. % S Furlow clear all % Load raw data from file filename=input('Enter complete file name of data file (no header)'); data=load(filename); data_end=15000; % Assign data more meaningful names vel=data(:,14); d_141=data(:,12); d_140=data(:,11); d_139=data(:,10); d_138=data(:,9)/1000; d_124=data(:,8); d_123=data(:,7); d_115=data(:,6)/1000; d_113=data(:,5)/1000; d_112=data(:,4)/1000; sw_103=1; offset_133=input('Enter preload offset for 103 should be in kips (8.5) '); d_103=(data(:,13)/1000-offset_133)*sw_103+(1-sw_103)*(data(:,3)/1000*17.99876*2.9e7/0.125); d_101=data(:,2)/1000; t=data(:,1)+4.696; % Filter Data dt=t(2)-t(1); fs=1/dt; nyq=fs/2; wn=[10/nyq,4500/nyq];
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[B,A] = BUTTER(3,wn); d_101_f = filter(B,A,d_101); d_103_f = filter(B,A,d_103); d_112_f = filter(B,A,d_112); d_113_f = filter(B,A,d_113); d_115_f = filter(B,A,d_115); d_123_f = filter(B,A,d_123); d_124_f = filter(B,A,d_124); d_138_f = filter(B,A,d_138); d_139_f = filter(B,A,d_139); d_140_f = filter(B,A,d_140); d_141_f = filter(B,A,d_141); % Build Velocity Windowed data d_vel=250; j=1; vel_i=vel(1); for i=1:numel(vel),
if (vel(i)-vel_i)>d_vel, index(j)=i; j=j+1; vel_i=vel(i); end
end index_vel_end=j; for i=1:index_vel_end-2, vel_a(i)=mean(vel(index(i):index(i+1))); d_101_f_std(i)=std(d_101_f(index(i):index(i+1))); d_103_f_std(i)=std(d_103_f(index(i):index(i+1))); d_112_f_std(i)=std(d_112_f(index(i):index(i+1))); d_113_f_std(i)=std(d_113_f(index(i):index(i+1))); d_115_f_std(i)=std(d_115_f(index(i):index(i+1))); d_123_f_std(i)=std(d_123_f(index(i):index(i+1))); d_124_f_std(i)=std(d_124_f(index(i):index(i+1))); d_138_f_std(i)=std(d_138_f(index(i):index(i+1))); d_139_f_std(i)=std(d_139_f(index(i):index(i+1))); d_140_f_std(i)=std(d_140_f(index(i):index(i+1))); d_141_f_std(i)=std(d_141_f(index(i):index(i+1))); dum_101=sort(abs(d_101_f(index(i):index(i+1))),'descend'); dum_103=sort(abs(d_103_f(index(i):index(i+1))),'descend');
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dum_112=sort(abs(d_112_f(index(i):index(i+1))),'descend'); dum_113=sort(abs(d_113_f(index(i):index(i+1))),'descend'); dum_115=sort(abs(d_115_f(index(i):index(i+1))),'descend'); dum_123=sort(abs(d_123_f(index(i):index(i+1))),'descend'); dum_124=sort(abs(d_124_f(index(i):index(i+1))),'descend'); dum_138=sort(abs(d_138_f(index(i):index(i+1))),'descend'); dum_139=sort(abs(d_139_f(index(i):index(i+1))),'descend'); dum_140=sort(abs(d_140_f(index(i):index(i+1))),'descend'); dum_141=sort(abs(d_141_f(index(i):index(i+1))),'descend'); d_101_f_p1(i)=dum_101(1); d_103_f_p1(i)=dum_103(1); d_112_f_p1(i)=dum_112(1); d_113_f_p1(i)=dum_113(1); d_115_f_p1(i)=dum_115(1); d_123_f_p1(i)=dum_123(1); d_124_f_p1(i)=dum_124(1); d_138_f_p1(i)=dum_138(1); d_139_f_p1(i)=dum_139(1); d_140_f_p1(i)=dum_140(1); d_141_f_p1(i)=dum_141(1); d_101_f_p2(i)=dum_101(2); d_103_f_p2(i)=dum_103(2); d_112_f_p2(i)=dum_112(2); d_113_f_p2(i)=dum_113(2); d_115_f_p2(i)=dum_115(2); d_123_f_p2(i)=dum_123(2); d_124_f_p2(i)=dum_124(2); d_138_f_p2(i)=dum_138(2); d_139_f_p2(i)=dum_139(2); d_140_f_p2(i)=dum_140(2); d_141_f_p2(i)=dum_141(2);
end
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% Frequency analysis [P_101_f,F_101] = PWELCH(d_101_f(1:8600),[],[],[],fs); [P_103_f,F_103] = PWELCH(d_103_f(1:13000),[],[],[],fs); [P_112_f,F_112] = PWELCH(d_112_f(1:13000),[],[],[],fs); [P_113_f,F_113] = PWELCH(d_113_f(1:13000),[],[],[],fs); [P_115_f,F_115] = PWELCH(d_115_f(1:13000),[],[],[],fs); [P_123_f,F_123] = PWELCH(d_123_f(1:13000),[],[],[],fs); [P_124_f,F_124] = PWELCH(d_124_f(1:13000),[],[],[],fs); [P_138_f,F_138] = PWELCH(d_138_f(1:13000),[],[],[],fs); [P_139_f,F_139] = PWELCH(d_139_f(1:13000),[],[],[],fs); [P_140_f,F_140] = PWELCH(d_140_f(1:13000),[],[],[],fs); [P_141_f,F_141] = PWELCH(d_141_f(1:13000),[],[],[],fs); % Export data fid = fopen('d_101_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_101,P_101_f]'); fid = fopen('d_103_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_103,P_103_f]'); fid = fopen('d_112_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_112,P_112_f]'); fid = fopen('d_113_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_113,P_113_f]'); fid = fopen('d_115_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_115,P_115_f]'); fid = fopen('d_123_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_123,P_123_f]'); fid = fopen('d_124_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_124,P_124_f]'); fid = fopen('d_138_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_138,P_138_f]'); fid = fopen('d_139_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_139,P_139_f]'); fid = fopen('d_140_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_140,P_140_f]');
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fid = fopen('d_141_psd_dads.txt','w'); fprintf(fid,'%6.4f %12.6f\n',[F_141,P_141_f]'); fid = fopen('d_101_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_101_f_std',d_101_f_p1',d_101_f_p2']'); fid = fopen('d_103_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_103_f_std',d_103_f_p1',d_103_f_p2']'); fid = fopen('d_112_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_112_f_std',d_112_f_p1',d_112_f_p2']'); fid = fopen('d_113_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_113_f_std',d_113_f_p1',d_113_f_p2']'); fid = fopen('d_115_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_115_f_std',d_115_f_p1',d_115_f_p2']'); fid = fopen('d_123_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_123_f_std',d_123_f_p1',d_123_f_p2']'); fid = fopen('d_124_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_124_f_std',d_124_f_p1',d_124_f_p2']'); fid = fopen('d_138_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_138_f_std',d_138_f_p1',d_138_f_p2']'); fid = fopen('d_139_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_139_f_std',d_139_f_p1',d_139_f_p2']'); fid = fopen('d_140_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f %12.6f\n',[vel_a',d_140_f_std',d_140_f_p1',d_140_f_p2']'); fid = fopen('d_141_f_v_dads.txt','w'); fprintf(fid,'%6.4f %12.6f %12.6f
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%12.6f\n',[vel_a',d_141_f_std',d_141_f_p1',d_141_f_p2']'); fid = fopen('d_101_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_101_f]'); fid = fopen('d_103_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_103_f]'); fid = fopen('d_112_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_112_f]'); fid = fopen('d_113_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_113_f]'); fid = fopen('d_115_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_115_f]'); fid = fopen('d_123_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_123_f]'); fid = fopen('d_124_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_124_f]'); fid = fopen('d_138_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_138_f]'); fid = fopen('d_139_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_139_f]'); fid = fopen('d_140_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_140_f]'); fid = fopen('d_141_f_dads.txt','w'); fprintf(fid,'%6.4f %6.4f %12.6f\n',[t,vel,d_141_f]'); fclose('all')
Published with MATLAB® 7.0
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data_plot.m Data Plotting Script for basic data overlay of DADS and Test
Data
clear all % Plot ? pl=1; % Variable array name=[101,103,112,113,115,123,124,138,139,140,141]; % Load and plot file data for i=1:numel(name), % Velocity Data for Plots of DADS and Mission filename=['d_',num2str(name(i)),'_f_dads.txt']; d1=load(filename); x2_d(:,i)=d1(:,2);y5_d(:,i)=d1(:,3); filename=['d_',num2str(name(i)),'_f.txt']; d2=load(filename); x2(:,i)=d2(:,2);y5(:,i)=d2(:,3); % Mean and STD Data for Plots of DADS and Mission filename=['d_',num2str(name(i)),'_psd_dads.txt']; d3=load(filename); x_d(:,i)=d3(:,1);y_d(:,i)=d3(:,2); filename=['d_',num2str(name(i)),'_psd.txt']; d4=load(filename); x(:,i)=d4(:,1);y(:,i)=d4(:,2); % PSD data for Plots of DADS and Mission filename=['d_',num2str(name(i)),'_f_v_dads.txt']; d5=load(filename); x1_d(:,i)=d5(:,1);y1_d(:,i)=d5(:,2); y2_d(:,i)=d5(:,3);y3_d(:,i)=d5(:,4); filename=['d_',num2str(name(i)),'_f_v.txt']; d6=load(filename); x1(:,i)=d6(:,1);y1(:,i)=d6(:,2); y2(:,i)=d6(:,3);y3(:,i)=d6(:,4); fclose('all');
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if pl,
figure,subplot(2,1,1), plot(x2_d(:,i),y5_d(:,i),'b',x2(:,i),y5(:,i),'r'),legend('DADS','Mission data'), xlabel('Velocity (fps)'), ylabel('Unit'),title(num2str(name(i))); hold on, subplot(2,1,2),plot(x1_d(:,i),y1_d(:,i),'b',x1_d(:,i),y2_d(:,i),'b+',x1_d(:,i),y3_d(:,i),'b+',x1(:,i),y1(:,i),'r',x1(:,i),y2(:,i),'r+',x1(:,i),y3(:,i),'r+'), legend('DADS STD','DADS Max','DADS Min','Mission data','Mission Max','Mission Min'), xlabel('Vel (fps)'), ylabel('Unit'),title(num2str(name(i))); figure,plot(x_d(:,i),y_d(:,i),'b',x(:,i),y(:,i),'r'), legend('DADS','Mission data'), xlabel('Freq (Hz)'), ylabel('Unit^2/Hz'),title(num2str(name(i)));
end clear x clear y clear x_d clear y_d end
Published with MATLAB® 7.0
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APPENDIX D
MATLAB PARAMETER VARIATION DATA PROCESSING
SCRIPTS
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param_proc.m Parameter Variation Data Processing Function
function []=param_proc(filename,plot_out) % param_proc(filename,plot) where filename is a string % and must be entered using single quotes --> 'filename' % plot --> plot option % This function is designed to process data from the % parameter variations filtering, then rearranging the % data into velocity windowed format % Finally the data is written into *.txt files. % This script does not perform any data decimation % S Furlow if filename=='2222222', dads_head_off(filename); else dads_head_off_dat(filename); end if filename(1,7)=='2', effective_weight=719.043 % sled weight (lbf) being acted upon by eta forces elseif filename(1,7)=='3', effective_weight=1177.863 % sled weight (lbf) being acted upon by eta forces elseif filename(1,7)=='1', effective_weight=260.213 % sled weight (lbf) being acted upon by eta forces else disp('Error in file input: file not recognized') end % Open no header file and process data data=load(strcat(filename,'_nh.dat')); % Assign data more meaningful names t=data(:,1); velocity=data(:,2); fy_s_aft_rt_in=data(:,3); fy_s_aft_rt_out=data(:,4); fy_s_aft_lt_out=data(:,5);
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fy_s_aft_lt_in=data(:,5); fy_s_aft_lt_in=data(:,6); fy_s_frt_rt_in=data(:,7); fy_s_frt_rt_out=data(:,8); fy_s_frt_lt_out=data(:,9); fy_s_frt_lt_in=data(:,10); fx_s_frt_rt_out=data(:,11); fx_s_frt_rt_in=data(:,12); fx_s_frt_lt_out=data(:,13); fx_s_frt_lt_in=data(:,14); fx_s_aft_rt_out=data(:,15); fx_s_aft_rt_in=data(:,16); fx_s_aft_lt_out=data(:,17); fx_s_aft_lt_in=data(:,18); m_xx=data(:,19); m_yy=data(:,20); m_zz=data(:,21); % Filter Data dt=t(2)-t(1); fs=1/dt; nyq=fs/2; wn=[10/nyq,0.9]; [B,A] = BUTTER(3,wn); fy_s_aft_rt_in_f= filter(B,A,fy_s_aft_rt_in);%fy_s_aft_rt_in fy_s_aft_rt_out_f = filter(B,A,fy_s_aft_rt_out);%fy_s_aft_rt_out fy_s_aft_lt_out_f = filter(B,A,fy_s_aft_lt_out);%fy_s_aft_lt_out fy_s_aft_lt_in_f = filter(B,A,fy_s_aft_lt_in);%fy_s_aft_lt_in fy_s_frt_rt_in_f = filter(B,A,fy_s_frt_rt_in);%fy_s_frt_rt_in fy_s_frt_rt_out_f = filter(B,A,fy_s_frt_rt_out);%fy_s_frt_rt_out fy_s_frt_lt_out_f = filter(B,A,fy_s_frt_lt_out);%fy_s_frt_lt_out fy_s_frt_lt_in_f = filter(B,A,fy_s_frt_lt_in);%fy_s_frt_lt_in fx_s_frt_rt_out_f = filter(B,A,fx_s_frt_rt_out);%fx_s_frt_rt_out fx_s_frt_rt_in_f = filter(B,A,fx_s_frt_rt_in);%fx_s_frt_rt_in
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fx_s_frt_lt_out_f = filter(B,A,fx_s_frt_lt_out);%fx_s_frt_lt_out fx_s_frt_lt_in_f = filter(B,A,fx_s_frt_lt_in);%fx_s_frt_lt_in fx_s_aft_rt_out_f = filter(B,A,fx_s_aft_rt_out);%fx_s_aft_rt_out fx_s_aft_rt_in_f = filter(B,A,fx_s_aft_rt_in);%fx_s_aft_rt_in fx_s_aft_lt_out_f = filter(B,A,fx_s_aft_lt_out);%fx_s_aft_lt_out fx_s_aft_lt_in_f = filter(B,A,fx_s_aft_lt_in);%fx_s_aft_lt_in m_xx_f = filter(B,A,m_xx);%m_xx m_yy_f = filter(B,A,m_yy);%m_yy m_zz_f = filter(B,A,m_zz);%m_zz % Construct Slipper reaction forces % Vertical Aft fy_s_aft_rt=fy_s_aft_rt_in_f+fy_s_aft_rt_out_f; fy_s_aft_lt=fy_s_aft_lt_out_f+fy_s_aft_lt_in_f; fy_s_aft=fy_s_aft_rt+fy_s_aft_lt; % Vertical Front fy_s_frt_rt=fy_s_frt_rt_in_f+fy_s_frt_rt_out_f; fy_s_frt_lt=fy_s_frt_lt_out_f+fy_s_frt_lt_in_f; fy_s_frt=fy_s_frt_rt+fy_s_frt_lt; % Vertical total fy_s=fy_s_aft+fy_s_frt; % Lateral Front fx_s_frt_rt=fx_s_frt_rt_out_f+fx_s_frt_rt_in_f; fx_s_frt_lt=fx_s_frt_lt_out_f+fx_s_frt_lt_in_f; fx_s_frt=fx_s_frt_rt+fx_s_frt_lt; % Lateral Aft fx_s_aft_rt=fx_s_aft_rt_out_f+fx_s_aft_rt_in_f; fx_s_aft_lt=fx_s_aft_lt_out_f+fx_s_aft_lt_in_f; fx_s_aft=fx_s_aft_rt+fx_s_aft_lt; % Lateral Total
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fx_s=fx_s_frt+fx_s_aft; % eta formation % Sled being acted upon by eta consists of sled body and % slipper beams and weighs % Vertical eta_v=fy_s./effective_weight; eta_l=fx_s./effective_weight; % Build Velocity Windowed data d_vel=250; j=1; vel_i=velocity(1); index(j)=1; for i=1:numel(velocity), if (velocity(i)-vel_i)>=d_vel, j=j+1; index(j)=i; vel_i=velocity(i); end end index_vel_end=j; for i=1:index_vel_end-1, vel_a(i)=mean(velocity(index(i):index(i+1))); eta_v_std(i)=3*std(eta_v(index(i):index(i+1))); eta_l_std(i)=3*std(eta_l(index(i):index(i+1))); m_xx_f_std(i)=3*std(m_xx_f(index(i):index(i+1))); m_yy_f_std(i)=3*std(m_yy_f(index(i):index(i+1))); m_zz_f_std(i)=3*std(m_zz_f(index(i):index(i+1))); eta_v_p(i)=max(abs(eta_v(index(i):index(i+1)))); eta_l_p(i)=max(abs(eta_l(index(i):index(i+1)))); m_xx_f_p(i)=max(abs(m_xx_f(index(i):index(i+1)))); m_yy_f_p(i)=max(abs(m_yy_f(index(i):index(i+1)))); m_zz_f_p(i)=max(abs(m_zz_f(index(i):index(i+1)))); end % Build envelope curve for data % Build on vertical data first
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j=1; peak_eta_v(1)=0; % force the zero-zero condition at zero velocity peak_vel_v(1)=0; index(1)=1; dum_slope=0; eta_v_p(1)=mean([0.0000,eta_v_p(2)]); % this step removes the artifical peak at low velocity for i=2:numel(eta_v_p), if slope(eta_v_p(i),vel_a(i),eta_v_p(index(j)),vel_a(index(j)))>=dum_slope, j=j+1; index(j)=i; peak_eta_v(j)=eta_v_p(i); peak_vel_v(j)=vel_a(i); end end if eta_v_p(i)<peak_eta_v(j), peak_eta_v(j+1)=peak_eta_v(j); % Plateau for the remainder of the velocity peak_vel_v(j+1)=vel_a(i); end % Build on lateral data next j=1; peak_eta_l(1)=0; % force the zero-zero condition at zero velocity peak_vel_l(1)=0; index(1)=1; dum_slope=0; eta_l_p(1)=mean([0.0000,eta_l_p(2)]); % this step removes the artifical peak at low velocity for i=2:numel(eta_l_p), if slope(eta_l_p(i),vel_a(i),eta_l_p(index(j)),vel_a(index(j)))>=dum_slope, j=j+1; index(j)=i; peak_eta_l(j)=eta_l_p(i); peak_vel_l(j)=vel_a(i); end end if eta_l_p(i)<peak_eta_l(j), peak_eta_l(j+1)=peak_eta_l(j);
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% Plateau for the remainder of the velocity peak_vel_l(j+1)=vel_a(i); end % Write Piecewise linear envelope function based on peak % values to account for all velocity values as found in % vel_a j=2;y2v=peak_eta_v(j);,x2v=peak_vel_v(j);y1v=peak_eta_v(j-1);x1v=peak_vel_v(j-1);sl_v=slope(y2v,x2v,y1v,x1v) k=2;y2l=peak_eta_l(k);x2l=peak_vel_l(k);y1l=peak_eta_l(k-1);x1l=peak_vel_l(k-1);sl_l=slope(y2l,x2l,y1l,x1l); for i=1:numel(vel_a), if vel_a(i)==peak_vel_v(j)&i~=numel(vel_a), j=j+1; y2v=peak_eta_v(j);x2v=peak_vel_v(j); y1v=peak_eta_v(j-1);x1v=peak_vel_v(j-1); sl_v=slope(y2v,x2v,y1v,x1v); end if vel_a(i)==peak_vel_l(k)&i~=numel(vel_a), k=k+1; y2l=peak_eta_l(k);x2l=peak_vel_l(k); y1l=peak_eta_l(k-1);x1l=peak_vel_l(k-1); sl_l=slope(y2l,x2l,y1l,x1l); end eta_v_out(i)=sl_v*(vel_a(i)-x1v)+y1v; eta_l_out(i)=sl_l*(vel_a(i)-x1l)+y1l; end % Prepare data for output and plot against peak curves % and lambda, check % plot option variable eta_vel_out=vel_a; if plot_out, lam_v=.008*eta_vel_out; lam_l=0.6*.008*eta_vel_out; figure,plot(vel_a,eta_v_p,'b-',peak_vel_v,peak_eta_v,'r--') hold on,plot(eta_vel_out,eta_v_out,'k-',eta_vel_out,lam_v,'r+'),title(strcat(filename,'_Vertical')),xlabel('Velocity (fps)'),ylabel('Factor (G)') figure,plot(vel_a,eta_l_p,'b-',peak_vel_l,peak_eta_l,'r--') hold on,plot(eta_vel_out,eta_l_out,'k-',eta_vel_out,lam_l,'r+'),title(strcat(filename,'_Lateral')),xlabel('Velocity (fps)'),ylabel('Factor (G)')
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end % Output data to file sf=size(filename); for i=1:sf(1,2), num_index(1,i)=str2num(filename(1,i)); end fid=fopen(strcat(filename,'.vef'),'w'); fprintf(fid,'%f %f %f \n',[0.00000000;0.00000000;0.00000000]); fid2=fopen(strcat(filename,'.eta'),'w'); [param_value]=param_ident(num_index); write_out=[param_value(1,1);param_value(1,2);param_value(1,3);param_value(1,4);param_value(1,5);param_value(1,6);param_value(1,7);0.00000000;0.00000000;0.00000000;0.00000000;0.00000000]; fprintf(fid2,'%f %f %f %f %f %f %f %f %f %f %f %f \n',write_out); dd=size(eta_vel_out) for j=1:dd(1,1), fprintf(fid,'%f %f %f \n',[eta_vel_out(j,:);eta_v_out(j,:);eta_v_out(j,:)*effective_weight]); end fclose(fid); for j=1:dd(1,2), write_out=[param_value(1,1);param_value(1,2);param_value(1,3);param_value(1,4);param_value(1,5);param_value(1,6);param_value(1,7);eta_vel_out(1,j);eta_v_out(1,j);eta_l_out(1,j);eta_v_out(1,j)*effective_weight;eta_l_out(1,j)*effective_weight]; fprintf(fid2,'%f %f %f %f %f %f %f %f %f %f %f %f \n',write_out); end fclose(fid2); j fid=fopen(strcat(filename,'.lef'),'w'); fprintf(fid,'%f %f %f \n',[0.00000000;0.00000000;0.00000000]); dd=size(eta_vel_out);
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for j=1:dd(1,1), fprintf(fid,'%f %f %f \n',[eta_vel_out(j,:);eta_l_out(j,:);eta_l_out(j,:)*effective_weight]); end
fclose(fid); Published with MATLAB® 7.0
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REFERENCES
846th Test Squadron (2006). SQUADRON OPERATING INSTRUCTION 10-3. Holloman AFB, NM: Author.
846th Test Squadron (2003). SRR FOREBODY FINAL ASSEMBLY, Engineering
Drawing Layout. Holloman AFB, NM: Author. Abbot, P. W. (1967). “Vibration Environment of the Holloman Impact Test Sled
(HIT) Second Stage”, ATM NO 348-44.1-6, 6585th Test Group Holloman AFB, NM.
Baker, W. E. and Turnbull, D. A. (1999). “Track Response to Simulated Slipper
Impacts” Report from Mixon Technologies Inc. under Contract #F08635-99-C-0023.
Cinnamon, J. D. (2006). “Analysis and Simulation of Hypervelocity Gouging
Impacts”, Doctoral Dissertation, Air Force Institute of Technology. Department of Defense. (2003). “Metallic Materials and Elements for Aerospace
Vehicle Structures: MIL-HDBK-5J” Department of Defense Handbook. Eskridge, S. E. (1998). “Development of Consistent Nonlinear Models of Flexible
Body Systems”, Master’s Thesis, Texas Tech University. Furlow, J. S. (2006a). "DADS Contact Formulation Using PRONTO FEM Results”,
Southern New Mexico Technology Symposium, 20 Apr 2006, Alamogordo, NM. Furlow, J. S. (2006b). "Dynamic Data Comparison of a Narrow Gauge Sled:
Numerical Data vs Test Data”, Southern New Mexico Technology Symposium, 20 Apr 2006, Alamogordo, NM .
Furlow, J. S. (2004). “DADS Flexible Body Formulation Study” Engineering
Seminar, Holloman High Speed Test Track, Oct 2004.
234
General Plastics Manufacturing Company (2000). Shock Mitigation Using Last-A-Foam High Density Flexible Foams. Tacoma, WA: Author.
Graf, K. F., Mahig, J., Wu, T. S., Barnes, R. A., Kowal, C. R. (1966). “Study of Track
Failure Criteria”, Report from The Ohio State University Research Foundation under USAF Contract No. AF 29(600)-5178.
Greenbaum, G. A., Garner, T. N., and Platus, D. L. (1973). “Development of Sled
Structural Design Procedures”, AFSWC-TR-73-22, 6585th Test Group, Holloman AFB, NM.
Gruttmann, F. and Wagner W. (2001). Shear Correction Factors in Timoshenko’s
Beam Theory for Arbitrary Shaped Cross-Sections, Computational Mechanics, 27, 199-207.
Haug, E. J. (1989). Computer Aided Kinetics and Dynamics of Mechanical Systems,
Volume I: Basic Methods. Boston: Allyn and Bacon. Herakovich, C. T. (1998). Mechanics of Fibrous Composites. New York: John Wiley
and Sons Inc. Hooser, C. D. (1989). “Narrow-Gage Rocket Sled Design Criteria Investigation”,
Master’s Thesis, New Mexico State University, 1989. Hooser, M. D. (2002a). “Rail and Deflection During the 80X-F1 HUP Sled Run”,
Holloman High Speed Test Track Technical Report, 12 June 2002. Hooser, M. D. (2002b). “The Holloman High Speed Test Track Gone Soft: Recent
Advances in Hypersonic Test Track Vibration Environment”, 21st AIAA Aerodynamic Measurement Technology and Ground Testing Conference, St. Louis, MO, AIAA 2002-3035.
Hooser, M. D. (2000). “Simulation of a 10,000 Foot per Second Ground Vehicle”,
21st AIAA Advanced Measurement Technology and Ground Testing Conference, 19-22 June 2000, Denver, CO, AIAA 2000-2290.
235
Hooser, M. D. and Mixon L. C. (2000). “Comparison of Sled Impact Momentum and Rail Momentum”, Holloman High Speed Test Track Technical Report, 1 March 2000.
Hooser, M. D. and Mixon L. C. (1998). “Rail Modal Tests”, Holloman High Speed
Test Track Technical Report, 6 April 1998. Hooser, M. D. and Schwing, A. (2000). “Validation of Dynamic Simulation
Techniques at the Holloman High Speed Test Track”, 38th Aerospace Sciences and Meeting Exhibit, 10-13 January 2000, Reno, NV, AIAA 2000-0155.
ISTRACON Handbook, (1961). Istracon Report No. 60-1, Interstation Supersonic
Track Conference, Structures Working Group, Air Force Systems Command. Holloman Air Force Base: Printing & Publications Distribution Division, AFMDC.
Krupovage, D. J., Mixon, L. C., and Bush, J. D. (1985). “Design Manual for Dual
Rail, Narrow Gage, and Monorail Rocket Sleds”, Holloman High Speed Test Track.
Krupovage, D. J., Mixon, L. C., and Bush, J. D. (1991). “Design Manual for Dual
Rail, Narrow Gage, and Monorail Rocket Sleds Revised”, Holloman High Speed Test Track.
Leadingham M. A. and Schauer S. A. (2001). Data Report for the Hypersonic
Upgrade Program (HUP) Super Roadrunner Over/Under (SRR O/U), 08 February 2001.
LMS CADSI. (1999). DADS/Flex Manual. Coralville, IA: Author. LMS CADSI. (2004). DADS Reference Manual. Coralville, IA: Author. Minto, D. W. (2004). “CTEIP Funded Advances in Hypersonic Testing at The
Holloman High Speed Test Track”, 24th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, Portland, OR, AIAA-2004-2740.
236
Minto, D. W. (2002). “The Holloman High Speed Test Track Hypersonic Upgrade Program Status”, 22nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference, St. Louis, MO, AIAA-2002-3034.
Minto, D. W. (2000). “Recent Increase in Hypersonic Test Capabilities at the
Holloman High Speed Test Track”, 38th Aerospace Sciences Meeting & Exhibit, Reno, NV, AIAA-2000-0154.
Mixon, L. C. (2006). Personal Communication. Holloman AFB, NM. 8 August 2006. Mixon, L. C. (2002a). Technical Report: Independent Analysis of the 6 Pupfish
Pusher Lateral Isolation System (Contract# F08635-99-C-0023), 23 January 2002.
Mixon, L. C. (2002b). Technical Report: Super Roadrunner Composite Case Modal
Test Report (Contract# F08635-99-C-0023), 1 February 2002. Mixon, L. C. (1999a). Technical Report: Initial Nike and SRR DADS Modeling
Parameters (Contract# F08635-99-C-0023), 01 June 1999. Mixon, L. C. (1999b). Technical Report: Design for Flexibility and Not Strength
(Contract# F08635-99-C-0023), 01 September 1999. Mixon, L. C. (1999c). Technical Report: Draft Loads Study on the 6 Pupfish Narrow
Gage Pusher Sled (Contract# F08635-99-C-0023), 01 December 1999. Mixon, L. C. (1971). “Sled Design Techniques”, AFSWC-TR-71-3, 6585th Test
Group, Holloman AFB, NM, Feb 1971. Mixon, L. C. and Hooser, C. D. (2002). Technical Report: PAC3 Long Body Program
Failure Analysis (Contract# F08635-99-C-0023), 2 September 2002. Nedimovic, P. H. (2004). “Dynamic Load Analysis for Design”, Masters Project
Report, New Mexico State University.
237
Przemieniecki, J. S. (1968). Theory of Matrix Structural Analysis. New York: McGraw Hill.
Szmerekovsky, A. G. (2004). “The Physical Understanding of the Use of Coatings to
Mitigate Hypervelocity Gouging Considering Real Test Sled Dimensions”, Doctoral Dissertation, Air Force Institute of Technology.
Stronge, W. J. (2000). Impact Mechanics. Cambridge: Cambridge University Press. Timoshenko, S. P., Young, D. H. and Weaver, W. (1974). Vibration Problems in
Engineering. New York John Wiley and Sons. Tischler, V. A., Venkayya, V. B., and Palazotto, A. N. (1981). “Dynamic Analysis of
High Speed Rocekt Sleds”, ASME Design Engineering Technical Conference, Hartford, CN, 81-DET-42.
Turnbull D. A. and Minto D. W. (2003). Data Report for the Hypersonic Upgrade
Program (HUP) Super Roadrunner Over/Under (SRR O/U) Record Sled Land-Speed & Payload-Energy Test, 20 May 2003.
Unigraphics Solutions Inc. (2002) I-DEAS User's Guide, Plano, TX: Author. White, F. M. (1994). Fluid Mechanics (3rd ed.). New York: McGraw-Hill, Inc. Volk, W. (1958). Applied Statistics for Engineers. New York: McGraw-Hill Book
Company, Inc. Zukas, J. A., Nicholas, T., Swift, H. F., Greszczuk, L, B. (1982). Impact Dynamics.
New York: John Wiley and Sons.
